Keywords: Monte Carlo, particle transport, kinetics, correlations, uctuations

Une fois arrivé à la n de la thèse (et surtout, à la n du manuscrit), il est tout de même satisfaisant de regarder derrière soi, et de pouvoir prendre dans les mains ce manuscrit tant craint, si plein d'imperfections, et dont on nit tout de même par être er. Ces trois années ont été parfois diciles, parfois grisantes, et surtout extrêmement remplies. Dans tous les cas, je n'aurais pas pu les aronter sans le soutien inconditionel de tant de gens, tant au SERMA que dans ma vie personnelle, et que j'aimerais remercier ici.

First, I would like to thank the members of my committee, Cheikh Diop, Sandra Dulla, Eric Dumonteil, and Alberto Rosso for taking time from their busy schedule to attend my viva in person. In particular, I would like to thank Eugene Shwageraus and Gaël Poette

Approximate results can be rapidly obtained, but their validity must be veried against reference calculations. Monte Carlo simulations, on the other hand, introduce almost no modelling errors, and thus play the role of references.

Monte Carlo simulations consist in sampling a large number of random trajectories of neutrons following probability laws derived from the underlying physical laws. Monte Carlo methods are exact but slow, because the statistical uncertainty is inversely proportional to the square root of the number of simulated histories. The statistical uncertainties can be reduced by modifying the sampling laws so that the average behaviour is preserved while the variance is modied. We thus speak of variance-reduction methods. Until recently, Monte Carlo simulations have only been applied only to stationary problems, due to their extremely high computational cost for nonstationary problems. In fact, the time-scale separation between the neutron lifetime (which is very short) and the precursor decay time (leading to further neutrons that are crucial for the reactor dynamics) represents a hard challenge for numerical simulation. However, recent technical and algorithmic advances have changed the status quo, enabling the use of Monte Carlo simulation for kinetics.

Variance-reduction methods aect all higher order statistical moments. Recent work has high-lighted peculiar phenomena in kinetic simulations, originating from space-time correlations between neutrons belonging to the same ssion chain. For instance, the clustering phenomenon describes the emergence of spontaneous neutron aggregation mechanisms, which are ultimately responsible for the appearance of a wild patchiness in the spatial neutron density.

In this thesis, we characterize the interaction between variance reduction methods and spacetime correlations in kinetic Monte Carlo methods, as well as the eects of this interaction on the reliability of uncertainty estimation for this class of simulations.

For this purpose, this thesis is divided in two parts. First, we consider a simplied model of a nuclear reactor, described by diusion theory.

We obtain analytical solutions for the pair correlation function describing correlations between pairs of particles, as well as for other observables of interest, such as the distribution of neutron families within ssion chains. Our investigation covers both the case of the free evolution of the neutron population, and the case of the evolution under constraint, which mimics the eect of population control on the neutron and precursor dynamics.

In the second part of the manuscript, we introduce a more realistic model, representative of the behaviour of kinetic Monte Carlo simulations realized in reference particle-transport codes such as TRIPOLI-4®, developed at CEA. This investigation helps to evaluate the eects of several Monte Carlo variance-reduction and populationcontrol algorithms on neutron correlations, with the aim of devising an optimal simulation strategy for quenching correlations, in stationary and non-stationary simulations.

Table des matières

2.2

Diagram illustrating the eects of Russian roulette and splitting . . . . . . . . . . . . . .

2.3

Example of use of weight combing for an initial population of N = 6 particles combed to M = 4 ; particles 2, 3, 4, 5 are copied once, and particle 1 and 6 are ignored. The nal tooth falls outside the particle bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Power iteration scheme when using ergodic average, with D inactive generations and G

3.2

Pair correlation function for a critical population starting with, on average, N = 100 prompt neutrons and M = 10 3 precursors, for a deterministic total population size N = N + M . Parameters are taken as in Table 3.1 for θ = 10 -1 . Blue solid line : analytical results from Eq. (3.35). Black circles : Monte Carlo simulations result with 10 6 replicas. .

3.4

Mean-squared pair distance in a one-dimensional box [-L, L] for critical populations using two sets of parameters taken from Table 3.1. Blue (gray) circles for Monte Carlo simulation with θ = 10 -2 ; blue (gray) solid line for analytical integration of Eq. (3.35) ; black triangles for Monte Carlo simulation with θ = 1 ; black pointed-line for the analytical integration of Eq. (3.10)

4.3

A typical history for a system of 4 neutrons and 1 precursor in the N -control model.

Neutrons diuse (in red line) until they branch or are destroyed by another neutron being produced in the system (these correlated event are shown in vertical blue dashed lines), and precursor lifetime is shown by horizontal black dashed lines. . . . . . . . . . . . . . .

4.4

Spatial dependence of the dierence between the pair correlation function evaluated in (x, y = -0.5) and (x = -L = -1, y = -0.5). Black circles : Monte Carlo simulation of the N -control scheme ; blue solid line : analytical solution of the anarchic model. Here, θ = 10 -3 . Parameters are taken from Table 3.1. . . . . . . . . . . . . . . . . . . . . . .

4.5

A typical history for a system of 2 neutrons and 1 precursor in the N M -control model.

Neutrons diuse (in red solid lines) until they branch or are destroyed by another neutron being produced in the system (these correlated event are shown in vertical blue dashed lines), and precursor lifetime is shown by horizontal black dashed-lines. . . . . . . . . . .

4.6

Asymptotic pair distance for dierent value of θ. The parameters are taken from Table 3.1. Black triangles : Monte Carlo simulations for the N -control scheme. Blue circles : Monte Carlo simulations of the N M -control scheme, both with 10 6 replicas. Blue solid line : analytical solution of the N M -control scheme Eqs.

4.9

A typical history for a system with 3 neutrons at t = 0, in the immigration model.

Neutrons diuse (in red) until they branch or are killed by another neutron branching somewhere else (vertical blue dashed lines show these correlated events). They can also be produced by the source, thus killing another randomly chosen neutron. . . . . . . . .

5.4

Shannon entropy as a function of the generations in power iteration, computed for N = 10 3 , B = 10 2 , and for dierent systems, and averaged over 10 3 independent replicas. . .

5.5

Asymptotic Shannon entropy S ∞ as a function of the bin size δ after discarding an adequate number of generations (cf.

5.6

Square root of pair distance as a function of the generations, computed for N = 10 3 , for dierent systems, with 10 3 independent replicas. . . . . . . . . . . . . . . . . . . . . . .

5.7

Average number of surviving families as a function of the generations, computed for N = 10 3 , for dierent systems, with 10 3 independent replicas. . . . . . . . . . . . . . .

5.8

Equilibrium Feynman moment for the ssion source computed for N = 1000 and δ = 0.5 for dierent systems.

5.16

Equilibrium normalized variance for the collision counts of neutrons, computed for N = 1000, δ = 0.5, and with 10 4 independent replicas, for the two heterogeneous systems. . . 5.17 Space-averaged equilibrium normalized variance for the collision counts of neutrons as a function of bin size, computed for N = 1000 and B = 100 for dierent systems. . . . . .

5.18

Shannon entropy of the ssion source, and the number of distinct families for a critical continuous energy box, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and = 8 3 is also provided. . . . . . . . . . . . . .

6.1

Shannon entropy as a function of time. The solid lines represent Monte Carlo results, and dashed black lines the ideal Shannon entropy computed for dierent population sizes. . .

6.2

Feynman moment for the neutron positions in the supercritical one-speed conguration, with initial population size N 0 = 250, at dierent times. . . . . . . . . . . . . . . . . . .

6.3

Feynman moment for the neutron positions in the subcritical one-speed conguration, with initial population size N 0 = 1000, at dierent times. . . . . . . . . . . . . . . . . .

6.4

Shannon entropy as a function of time when applying weight combing at each time step.

The solid lines represent Monte Carlo results, and the pointed lines the ideal Shannon entropy computed for the total neutron population at a given time. . . . . . . . . . . . .

6.5

Feynman moment for the neutron positions in the supercritical one-speed conguration, with initial population size N 0 = 250, at dierent times. . . . . . . . . . . . . . . . . . .

6.6

Feynman moment for the equilibrium source in the supercritical one-speed conguration, with L = 400, with initial population size N 0 = 250, at dierent times. . . . . . . . . . .

6.7

Entropy of the neutron positions as a function of time. Blue line : without combing ; orange line : with combing at each time step and on the equilibrium source. . . . . . . .

6.8

Feynman moment of the neutron positions in the supercritical small system with leakage,

6.29

Feynman moments of the collision density at dierent times, with a total population N = 10 4 and M = 10 4 independent replicas. Squares are for the fast group, circles for the epithermal group, and triangles for the thermal group. Blue is for t = 0.001 s ; red is for t = 0.1 s ; orange is for t = 1 s, and green for t = 10 s. . . . . . . . . . . . . . . . . 6.30 Feynman moments of the precursor positions at dierent times for system 2 in stationary conditions. The total population is N = 10 4 and the number of independent replicas is M = 10 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.31 Shannon entropy of the neutron and precursor populations when changing reactivity at t = 0, with a total population N = 10 4 and M = 10 4 independent replicas. Blue for system 1 and red for system 2 ; solid lines for the reactivity removal, and cross markers for the reactivity insertion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.32

Feynman moments of the collision density at dierent times, with a total population N = 10 4 and M = 10 4 independent replicas. We consider system 2 for the supercritical and for the subcritical transients. Squares are for the fast group, circles for the epithermal group, and triangles for the thermal group. Blue is for t = 0.001 s ; red is for t = 0.1 s ; orange is for t = 1 s, and green for t = 10 s. . . . . . . . . . . . . . . . . . . . . . . . . Dierent sets of parameters used in this work. Here N is the initial (average) number of neutrons, M is the initial (average) number of precursors, L is the box half-size, D is the diusion coecient, β is the ssion rate, γ is the capture rate, λ is the decay rate of precursors, ν p,l and ν d,l are the l-th falling factorial moments of the number of neutrons and precursors produced from ssion, respectively, and θ = λ/(βν d,1 ). Units are arbitrary.

5.3

Nuclear data of the UO 2 material. The three energy groups are indexed by g. . . . . . .

5.4

Nuclear data for the H 2 O material. The three energy groups are indexed by g. . . . . . .

5.5

Capture sections Σ c,g for each conguration. The three energy groups are indexed by g. .

5.6

Isotopic composition of the reected continuous-energy cube reactor. . . . . . . . . . . .

6.1

Parameters for the one-speed problem with vacuum boundary conditions . . . . . . . . .

6.2

Fission yields for the congurations with delayed neutrons, and corresponding multiplication factors and leakage probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Nuclear data for the precursors in the homogeneous material. . . . . . . . . . . . . . . .

6.4

Capture cross sections for the reactivity insertions in system 1, along with the new prompt ssion spectrum. The energy groups are indexed by g. . . . . . . . . . . . . . . . . . . .

6.5

Capture cross sections for the reactivity insertions in system 2. The energy groups are indexed by g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6

Amplitude of the reactivity insertions for each case. . . . . . . . . . . . . . . . . . . . .

6.7

Expected fractional weight in the precursor source. The families are indexed by j. . . . .

-Introduction

In the context of nuclear reactor safety, the development of predictive, reliable and fast simulation tools for multi-physics simulations coupling neutron ux solvers with thermal-hydraulics and thermomechanics codes, under stationary and transient conditions, is the subject of very extensive research programs in several leading institutions 1 .

The neutron density, i.e. the average number of particles in phase space 2 , obeys the Boltzmann equation [2]. Analytical solutions for this equation are beyond reach, except for highly simplied congurations ; numerical simulations are thus mandatory. Thanks to the linearity of the Boltzmann equation and to the intermediate dimension of the phase space (neither too high nor too low), deterministic and Monte Carlo methods can be both successfully used to obtain numerical solutions, each presenting specic advantages and drawbacks.

The Boltzmann equation in non-stationary conditions is traditionally solved using deterministic methods, which rely on the discretization of the phase space. Because of the extremely large number of unknowns (of the order of ∼ 10 14 ) that would result from a reasonably ne discretization of space, direction and time variables, most state-of-the-art deterministic simulation codes adopt the so-called two-step approach, whereupon the results of a nely-discretized calculation at the scale of a fuel assembly (or a collection of fuel assemblies) in stationary conditions is used to prepare the homogenized and condensed nuclear data that are then injected into time-dependent simplied transport solvers with coarse energy discretization for simulations at the full-core scale [START_REF] Miao | Analysis of correlations and their impact on convergence rates in Monte Carlo eigenvalue simulations[END_REF]. Often, a factorization of the phase space is introduced, separating time with respect to the classical phase-space variables [START_REF] Dulla | The quasi-static method revisited[END_REF][START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF][START_REF] Dulla | Some new thoughts on the multipoint method for reactor physics applications[END_REF].

Deterministic solvers are computationally fast, but they achieve their speed at the expense of introducing discretization biases and reactor-dependent models (e.g., self-shielding) that call for the validation of this class of numerical tools against high-delity simulations and experimental measurements [START_REF]Safety Margins of Operating Reactors -Analysis of Uncertainties and Implications for Decision Making[END_REF].

Experimental data concerning nuclear reactors in the stationary state are generally available, although somewhat scarce since in most cases they belong to plant operators and are thus not directly accessible.

In this respect, remarkable initiatives aimed at establishing international benchmarks for code validation (including depletion and multi-physics coupling) have been recently promoted, such as VERA (Virtual Environment for Reactor Applications) 3 , BEAVRS (Benchmark for Evaluation And Validation of Reac- tor Simulations) 4 , and Watts Bar (TVA Watts Bar Unit 1 Multi-Physics Benchmark) 5 . The situation is denitely worse for data pertaining to non-stationary regimes, such as operational transients following the insertion of control rods, or experimental campaigns aimed at probing the reactor behaviour under external solicitations, which are seldom available. Extreme cases, such as accidental transients which might lead to fuel degradation, are dangerous and not directly accessible by experimental means, in spite of their relevance for the validation of simulation codes [START_REF]Research reactor benchmarking database: Facility specication and experimental data[END_REF].

In view of these considerations, Monte Carlo simulations play a special role, in that they do not suer from discretization errors [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF] and can thus provide a high-delity `golden standard' for the validation of deterministic solvers ; furthermore, they become de facto mandatory to assess the biases of deterministic solvers in non-stationary reactor regimes. 1 See for instance the initiatives promoted in the USA, such as the CESAR project (cesar.mcs.anl.gov) or the CASL consortium (www.casl.gov), and the European projects FP7 HPMC and McSAFE (cordis.europa.eu/ projects).

1.1 . Monte Carlo simulations : from stationary to time-dependent Monte Carlo simulations for particle transport problems are based on the random sampling of the particles' trajectories in the phase space. For the applications of interest in reactor physics, particles are primarily neutrons, delayed-neutron precursors and photons. In this thesis, we will focus on neutrons and precursors. Their random interactions with the surrounding media are sampled in accordance with the probability laws given in the nuclear data libraries ; particle tracking is determined by computing the intersections of their ights between collisions with the cells of the geometrical model. Trajectories are followed until particles disappear, either by capture events or by leaking from the boundaries of the system. Along each trajectory, physical observables such as reaction rates are estimated by setting appropriate `tallies' and collecting statistics (mean and variance) over a suciently large ensemble of sampled trajectories. The sample average over particle histories yields unbiased estimates of the expected value of the sought observables, from the law of large numbers ; the sample variance yields the associated statistical uncertainty of the sample average, from the central limit theorem [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. The variance of the sample average is inversely proportional to the number of sampled histories, which makes Monte Carlo simulations typically (much) slower than deterministic solvers. However, this limitation is compensated by the unbiasedness of the obtained estimates : deterministic and Monte Carlo codes complement each other and are equally necessary.

Mainly due to considerations related to the required simulation time, combined with limited computer power, in reactor physics Monte Carlo codes have been long applied almost exclusively to the solution of stationary problems in isothermal conditions. Recent advances in both algorithms and computer power have however paved the way towards a whole range of new applications for Monte Carlo simulations, specically aimed at extending the perimeter of applicability of these methods to multi-physics problems.

Enormous progress has been achieved e.g. in the eld of depletion calculations, where the behaviour of the neutron density is coupled to the transmutation of the media under irradiation, whose evolution is ruled by the generalized Bateman equations : Monte Carlo codes have been eciently coupled with (deterministic) solvers for the Bateman equations, and the stability of the resulting simulation schemes has been thoroughly investigated [1116].

Over the last ten years, there has been a growing interest in the development of a novel class of Monte Carlo methods devoted to kinetic (i.e., time-dependent) simulations, whose computing time was prohibitively high until quite recently. Prompted by the pioneering work by Sjenitzer and Hoogenboom [START_REF] Sjenitzer | A Monte Carlo method for calculation of the dynamic behaviour of nuclear reactors[END_REF][START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF], ingenuous variance-reduction and population-control methods targeted at eciently handling timedependent problems have been rst tested on simplied congurations, and then successfully extended to realistic reactor models with thermal-hydraulics feedback eects [1926].

. Correlations and uctuations in kinetic Monte Carlo simulations

Due to the still relatively recent appearance of these breakthroughs, a comprehensive framework for the investigation of time-dependent Monte Carlo simulations for reactor physics applications is yet to be established, despite many promising early investigations. The accuracy and precision of Monte Carlo methods is crucial, because they are intended as a golden standard with respect to which simplied approaches are to be compared. In particular, the reliability of the sought results is only as good as the one of the associated condence intervals, which are customarily estimated by inference techniques, i.e. collecting the sample variance over Monte Carlo histories. The sample variance captures the uctuations in the tallies due to the inherently stochastic nature of particle histories. If the particle histories are independent, and the uctuations are well-behaved, then the quantication of the sample variance is straightforward. However, kinetic simulations pose peculiar challenges in this respect.

To begin with, the initial neutron and precursor source for kinetic problems often corresponds to the equilibrium (stationary) reactor condition, which means that the particles beginning the time-dependent simulation inherit the correlations that are known to aect the eigenvalue calculations needed to sample the equilibrium particle conguration [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. When the number of simulated particles is low, these correlations are enhanced, and manifest themselves in the form of spontaneous aggregation of particles (`clustering'), resulting in patchy spatial distributions [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF] such as those illustrated in Fig. 1.1. Furthermore, kinetic Monte Carlo simulations come with their own sets of problems : particle histories may `branch' because of ssion events inducing secondary neutrons, which increases the overall variance and is responsible for correlations in time and in space, even when the reactor is critical. These eects might in some extreme cases lead to an unbounded growth of the variance over time, dubbed critical catastrophe [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

In the following, we provide a concise overview of the state-of-the-art knowledge on these issues, the key open questions and their relevance for applications.

. Correlations in eigenvalue Monte Carlo simulations

As illustrated in the following chapters, the equilibrium neutron and precursor population can be sampled by Monte Carlo methods simulating ssion chains over several generations (a generation being dened as the neutron history between birth by ssion and death by leakage or absorption), until the stationary regime is attained [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. At each generation, the genealogy of any neutron can be thus traced back to an ancestor, and the set of neutrons sharing a common ancestor forms a family : events pertaining to the same family will be correlated. Such correlations aect both the statistics of a tally over generations at a given spatial cell, and the statistics of tallies at distinct spatial cells within a generation : these two points of view are intimately related.

The assessment of generational correlations is a long-lasting problem. In eigenvalue Monte Carlo simulations, the average and the variance are typically estimated over successive generations by assuming that tallies collected in each generation are identically and independently distributed. This corresponds to an ergodic averaging procedure, by analogy with the ergodic assumption in equilibrium statistical physics.

While the tallies are indeed identically distributed (after convergence has been reached), they are also correlated, in view of the previous considerations, and this aects the estimation of their uncertainty.

Brissenden and Garlick rst noted that neglecting generational correlations when estimating the variance using ergodic averaging leads to an underestimation of the variance of the tallies [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF] : one has thus to distinguish between the (underestimated) apparent variance obtained through ergodic averaging and the true variance that would have been obtained through ensemble averaging over a set of fully independent replicas 6 . Ueki et al. [START_REF] Ueki | Autocorrelation and dominance ratio in Monte Carlo criticality calculations[END_REF] showed that the discrepancy between apparent and true variance may be signicant for local (i.e., spatially distributed) tallies. They also showed that the convergence rate of the apparent variance with respect to the number of simulated particles deviated from the scaling expected for independent histories, although later Herman demonstrated that the usual scaling is asymptotically recovered for large population sizes [START_REF] Herman | Monte Carlo and thermal hydraulic coupling using low-order nonlinear diusion acceleration[END_REF]. Min-Jae et al. found that the ratio of the apparent-to-true variance is not uniform in space [START_REF] Lee | Multigroup monte carlo reactor calculation with coarse mesh nite dierence formulation for real variance reduction[END_REF].

Methods aiming at predicting the true variance based on the apparent variance have been proposed, most often relying on simplifying assumptions : Yamamoto et al. modelled the underestimation of the variance of local tallies and its dependency on space and tallying bin size [START_REF] Yamamoto | Prediction on underestimation of variance for ssion rate distribution in monte-carlo calculation[END_REF]. Sutton applied a discretized phase-space approach to understand the spatial non-uniformity of the apparent-to-true variance ratio [START_REF] Sutton | Application of a discretized phase-space approach to the analysis of Monte Carlo uncertainties[END_REF]. Miao et al. computed the auto-correlation coecients between spatial tallies to analyse the behaviour of generational correlations [3537].

The impact of spatial correlations in eigenvalue Monte Carlo simulations was recently highlighted by Dumonteil et al., who rst pointed out the emergence of neutron clustering [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF] ; diagnostic tools to detect spatial correlations were later developed by Nowak et al. [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF] building upon previous ndings [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

Although such spatial correlations are not supposed to alter the average tallies, Cosgrove et al. have shown that a potentially more problematic situation arises when Monte Carlo simulations are coupled to a fuel depletion solver, which might amplify existing numerical instabilities [START_REF] Isotalo | Preventing xenon oscillations in Monte Carlo burnup calculations by enforcing equilibrium xenon distribution[END_REF][START_REF] Dufek | Numerical stability of the predictorcorrector method in Monte Carlo burnup calculations of critical reactors[END_REF] and induce biases on the average observables [START_REF] Cosgrove | Neutron clustering as a driver of Monte Carlo burn-up instability[END_REF]. Sutton has proposed to relate the behaviour of neutron clustering to the statistics of the families, observing that a smaller number of independent families (or a larger family size) induces stronger correlations [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF]. This research eld is still unsettled, and somewhat contradictory statements may be found in the literature. Some authors have suggested that clustering might have an impact also on the average tallies, beside aecting the estimation of the variance [START_REF] Dumonteil | Clustering and traveling waves in the Monte Carlo criticality simulation of decoupled and conned media[END_REF][START_REF] Fröhlicher | Generational variance reduction in monte carlo criticality simulations as a way of mitigating unwanted correlations[END_REF]. In stark contrast, other authors have suggested that spatial correlations have no eect on the estimation of statistical uncertainty [START_REF] Mickus | Does neutron clustering aect tally errors in Monte Carlo criticality calculations?[END_REF].

. Correlations in time-dependent Monte Carlo simulations

Time-dependent Monte Carlo simulations for reactor physics applications have only recently been introduced, which explains why the analysis of the behaviour of correlations in this class of problems is still in its infancy. Furthermore, despite eective advances in variance-reduction methods, the computing cost of kinetic problems is still very high, so that the investigation of correlations is most often carried out in simplied models with varying degrees of approximations.

Dumonteil et al. initially assessed spatial correlations at a given observation time by disregarding

the contributions of precursors and assuming unbounded media [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF]. The eects of bounded media on space-time correlations were later taken into account [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF], leading to the development of a more rened model where the neutron population is kept constant to mimic the impact of population-control methods [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. An interesting feature of these models is that most of their ndings carry over also to generation-based eigenvalue problems, if one replaces time by the discrete generation index [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF] : this also means that the open questions concerning the latter translate into open questions concerning the former. Unsurprisingly, the main nding is that clustering occurs due to ssion events, similarly as in the case of eigenvalue problems. If no population control is applied, the variance of a critical system will grow unbounded, eventually leading to an abnormal termination of the kinetic simulations (critical catastrophe) ; population control averts the critical catastrophe. The impact of precursors (in unbounded media, and in the absence of population control) has been examined by Houchmandzadeh et al. [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF],

who have shown that the amplitude of the correlations was dramatically quenched and their evolution in time considerably slowed down for systems close to criticality.

All these investigations are restricted to spatially homogeneous media and simplied transport models, which hinders the immediate applicability of these ndings to more realistic applications, although some relevant general tendencies can be successfully extracted. An even more serious limitation is that the models mentioned above rely on the hypotheses of analog Monte Carlo simulations, excluding variancereduction techniques. In view of its utmost importance for production Monte Carlo codes, the interference between non-analog Monte Carlo methods, variance-reduction and population-control techniques and the behaviour of correlations in kinetic problems calls for a deeper understanding, as witnessed by recent contributions [START_REF] Variansyah | Analysis of population control techniques for time-dependent and eigenvalue Monte Carlo neutron transport calculations[END_REF].

. Goal and structure of the thesis

Based on these considerations, in this thesis we set out to establish a unied and coherent framework for the analysis of uctuations and correlations in kinetic Monte Carlo simulations, with the goal of ensuring the reliability of the statistical uncertainty estimates in such calculations. In particular, we will focus on the non-trivial interplay between the behaviour of uctuations and correlations and the application of variance-reduction and population-control techniques, as well as sampling strategies used for non-analog Monte Carlo simulations. Our investigation will cover both algorithms devoted specically to time-dependent problems and algorithms devoted to (stationary) eigenvalue problems, by virtue of their central role for the sampling of the equilibrium neutron and precursor population. The results of our analysis and the novel approaches that we have chosen in order to achieve our goals will be presented in details in the remainder of this manuscript. This work is structured as follows :

• First, in Chapter 2, we present the general background on Monte Carlo simulations in reactor physics, with special emphasis on the methods used for kinetic problems.

• In Part I (Chapters 3 and 4), we shall present our results in a simplied diusion-based analog kinetic Monte Carlo simulation, where a population of neutrons and precursors evolves in time following a Branching Brownian Motion. We will characterize ssion-induced correlations in such a system and derive characteristic quantities in order to facilitate the analysis of more realistic simulations. A focus on the eects of an idealized population-control mechanism in analog simulations will be introduced.

• In Part II (Chapters 5 and 6), we shall present our results regarding non-analog kinetic Monte Carlo simulations, making use of a broad range of variance-reduction and population-control techniques. We will propose suitable estimators for the analysis of correlations in such systems, and we will discuss the eects of variance-reduction and population-control methods on uctuations and correlations.

In Chapter 2 we will briey recall the main concepts of neutron transport in reactor physics, in a general time-dependent setting of the Boltzmann equation, including the contributions of the delayedneutron precursors. Simplied formulations of the Boltzmann equations, encompassing the multi-group and diusion approximations, will be illustrated. Next, we will introduce the basic principles of Monte Carlo simulations, both analog and non-analog, to estimate the solution of the time-dependent Boltzmann equation ; we will in particular discuss the most important variance-reduction and population-control techniques that will be used in the remainder of the manuscript. Special cases of Monte Carlo algorithms to solve the multi-group and diusion versions of the Boltzmann equations will be detailed. Chapter 3 will introduce a diusion-based simplied kinetic Monte Carlo simulation scheme based on Branching Brownian Motion that includes delayed events in order to account for precursors. We will analyze ssion-induced spatial correlations between particles using the space-dependent moment equations, based on the Pál-Bell backward formalism. Within the framework of this simplied transport model, we will derive exact expressions for the typical length and time scales of the correlations in this class of simulations.

A step toward the characterization of population-control algorithms will be provided in Chapter 4 by enforcing idealized constraints on neutrons and precursors, still within the framework of diusion-based simplied kinetic Monte Carlo simulations. Relying on the forward formulation of the master equation, we will establish moment equations whenever possible for the particle statistics, and derive analytical (or semi-analytical) expressions for the key observables. Such quantities will be compared to Monte Carlo simulations, used as a reference, and to the results obtained without population control. In addition, we shall complement the analysis of the moment equations by a discussion on the statistical behaviour of neutron `families' dened over ssion chains.

In Chapter 5, we will turn to the more realistic case of non-analog Monte Carlo simulations in the framework of the multi-group Boltzmann equation. The rst step of our analysis will consist in examining the initial particle source corresponding to a critical reactor, which is often required in timedependent problems. This source depends on a preliminary stationary Monte Carlo simulation to solve the k-eigenvalue problem and access the fundamental eigenstate. The power iteration algorithm used to converge towards the fundamental mode is itself prone to non-trivial eects due to inter-generation and intra-generation correlations, which we will investigate extensively in view of their impact on the source for kinetic simulations. For this purpose, we will use a few benchmark congurations including spatially homogeneous and heterogeneous systems. We will assess the impact of various variance-reduction and population-control techniques on correlations in stationary Monte Carlo methods, relying on usual and novel tallies.

In Chapter 6, we will nally examine the case of non-analog kinetic Monte Carlo simulations, still within the framework of the multi-group Boltzmann equation. We will rst consider a simplied benchmark conguration that will allow extracting the key tendencies of correlations : in particular, we will examine the impact of precursors, which will be shown to play a key role in determining the amplitude and the spatial behaviour of the correlations. Then, we will revisit the more realistic benchmark congurations introduced in Chapter 5. We will assess the correlations of the equilibrium neutron and precursor populations, as determined by running the power iteration described in the previous chapter, and then analyze time-dependent simulations. Three dierent regimes will be explored : critical, subcritical and supercritical.

Conclusions will be nally drawn in Chapter 7, where we will also present the main perspectives for future work.

. Published works

Part of the material presented in this thesis has previously appeared in the following publications : • T. Bonnet, H. Belanger, D. Mancusi, and A. Zoia, "The eect of branchless collisions and population control on correlations in Monte Carlo power iteration", submitted to Nuclear Science and Engineering. Preprint avalaible : https://arxiv.org/abs/2309.03767 Furthermore, some of the results have been or will be presented (oral talks) at the following international conferences and workshops :

• T.
• International Conference on Transport Theory (ICTT), Bertinoro, Italy, Jul. 10-16th, 2022.

• M&C 2023 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Niagara Falls, Canada, Aug. 13-17th, 2023.

• Fourth annual workshop "Mathématiques pour la neutronique", GdR MaNu, INSTN/Saclay, France, Dec. 7th, 2022.

• Invited talk at the Inaugural meeting, GdR Branchement, Institut de Mathématiques de Toulouse, France, Nov. 8-10th, 2023.

-Time-dependent Monte Carlo simulation for reactor physics applications

In this chapter, we introduce the question of correlations in time-dependent neutron-transport Monte Carlo simulations. We start by recalling the basic features of the time-dependent Boltzmann equation, which governs neutron transport in the kinetic regime, and introducing notations and concepts that will be useful in the following. Then, we illustrate how Monte Carlo methods can be used in order to solve the time-dependent Boltzmann equation, and briey review the key variance-reduction and populationcontrol strategies that have been recently proposed to cope with the distinct challenges of kinetic Monte Carlo simulations.

. A brief introduction to neutron-transport problems

. Neutron-matter interactions

Neutrons are neutral particles that interact with matter only through collisions with the surrounding nuclei, and otherwise stream freely. Although the neutron is unstable, and spontaneously decays to a proton, the typical decay time in void is of the order of ten minutes, a timescale which is much longer than the typical neutron life-time within nuclear systems, i.e., shorter than 10 -2 s for the longest-lived particles.

For this reason, the spontaneous decay can be safely neglected for the applications to reactor physics that we will examine in the following. Since the energy range of interest for reactor physics is conventionally assumed to lie between 20 MeV and 10 -5 eV, relativistic eects at high energy can be neglected in most applications. Neutron-matter interactions should in principle be described using quantum mechanics, since the de Broglie wavelength of the neutron ranges between 10 -13 cm for neutrons at high energy (say at 20 MeV) and 10 -8 cm for neutrons that are in thermal equilibrium with the traversed medium (say at 0.025 meV) : these scales are comparable, respectively, with the typical radius of nuclei and with the typical distance over which molecular bonds develop. However, the common practice in reactor physics is that the quantum-physics aspects required for collision events are pre-computed by nuclear physics specialists and condensed in tabulated data that can be accessed and interpolated as needed, including interaction probabilities and the probability of selecting a specic reaction channel (e.g., elastic or inelastic scattering, capture or ssion), energy-angle distributions for the outgoing neutrons, and the associated yields. Thus, it can be safely assumed that neutrons evolve in a classical phase space with coordinates P = {r, Ω, E, t}, where r is the position vector, Ω is the direction vector, E is the energy, and t is time. Within this framework, the probability that an incident neutron interacts with a nucleus i in the medium via reaction channel k is assigned by introducing the notion of partial microscopic cross sections σ i,k (E), that carry units of an area (and are usually measured in units of barns, where 1 b = 10 -24 cm 2 ).

The microscopic cross sections have a highly intricate dependence on the energy E, which basically stems from the fact that they condense all the complex quantum-physics interaction laws, as mentioned earlier.

The sum over all the reaction channels k yields the total microscopic cross section for a given nuclide, namely,

σ i,t (E) = k σ i,k (E), (2.1)
so that p i,k (E) = σ i,k (E)/σ i,t (E)
expresses the probability of choosing the reaction channel k, conditioned on the event of having a collision with the nuclide i.

The allowed reactions channels are often grouped into three categories : scattering, sterile absorption, and ssion [2] :

• Scattering can be either elastic or inelastic, and most often leads to a unit yield ; reactions of the class (n, xn), with x = 2, 3 or even 4 are possible, albeit rarer in the energy range of interest for reactor physics. The energy-angle spectrum of the emerging neutrons is in most cases rather involved, and strongly depends on the nature of the collided nuclide. In the remainder of the manuscript, we will mainly focus on the simple case of isotropic scattering with unit yield.

• A neutron undergoing a capture (i.e., sterile absorption) stays within the collided nuclide and thus disappears with respect to the transport process : the neutron history is terminated. Other particles might be emitted, such as photons or charged particles, but here we will exclusively focus on neutrons, and such events will be neglected.

• Upon ssion, the incident neutron is absorbed and splits the target nucleus into (most often) two `fragments'. Contrary to sterile absorption, ssion leads to the emission of secondary neutrons. The number of emitted neutrons is random. On average, ν p so-called `prompt' neutrons of high energy (e.g. around 2 MeV for Uranium 235) are emitted almost immediately after the ssion event. It is reasonable to assume that the angular distribution of the prompt neutrons is isotropic. The shape of the energy spectrum χ p (E) can be approximated with simple analytical forms, such as the one proposed by Watt [START_REF] Watt | Energy spectrum of neutrons from thermal ssion of U235[END_REF]. The ssion fragments are left in excited states, which generally de-excite via β -decay. In some cases, additional neutrons may be emitted along the decay chain : since these neutrons are emitted over the timescale of β -decay (110 s for Uranium 235), they are dubbed `delayed neutrons'. Correspondingly, due to their role, such excited states take the name of delayed-neutron precursors. The number of created precursors per ssion event is random, with average ν d . The ssion fragments are heavy and lose almost all their kinetic energy within microns with respect to the ssion event : for this reason, it is assumed that delayed neutrons are emitted at the same location as the prompt neutrons originating from the same ssion event. The energy spectrum χ d (E) of delayed neutrons has a shape that is qualitatively similar to the one of prompt neutrons, but with a sensibly smaller average energy, of the order of 400 keV for Uranium 235.

The angular distribution can again be taken as isotropic. The total number of possible precursors is about 300 : for the sake of simplicity, in nuclear data libraries they have been grouped into a small number of `families' (6 or 8), each family j having spectrum χ j d (E), decay constant λ j and yield ν j d , with ν d = j ν j d . The ratio β = ν d /(ν p + ν d ) is small, of the order of 0.6% for Uranium 235.

In general, a medium is composed of many dierent nuclides, each having a total microscopic cross section σ i,t (E) and a concentration N i (r) (usually in units of nuclides per cubic centimeter). The probability of selecting a given nuclide upon a collision at r induced by a neutron of energy E is thus given by

p i (r, E) = N i (r)σ i,t (E) j N j (r)σ j,t (E)
.

(2.

2)

The interaction probability with the medium per unit ight length for a reaction k is then given by the partial macroscopic cross section Σ k , carrying units of inverse length (usually, cm -1 ), i.e.,

Σ k (r, E) = i N i (r)σ i,k (E).
(2.3)

Finally, the total interaction probability per unit ight length within a medium is given by the total macroscopic cross section

Σ t (r, E) = k Σ k (r, E).
(2.4)

It is assumed that the media of interest for reactor physics applications are suciently isotropic at the spatial scales seen by the neutrons, so that in practice the cross sections do not depend on the direction of ight Ω.

Since neutrons freely stream within the medium and only interact with nuclides upon collision, with

Poisson probability Σ t (r, E)ds for a ight of innitesimal length ds at position r, in direction Ω and with energy E, it follows that the ight lengths s between collision events are distributed according to a non-homogeneous exponential probability density f (s) with parameter Σ t (r, E), namely,

f (s) = Σ t (r + sΩ, E)e - s 0 Σ t (r + s ′ Ω, E)ds ′ .
(2.5)

When the material properties of the medium are spatially homogeneous, Eq. (2.5) becomes a simple exponential probability density function.

The overall picture of the physical mechanisms recalled here shows that the succession of ights and collisions forms a chain of stochastic events, with ssion playing a special role in that new `branches' might appear. For illustration, the stochastic dynamics of a representative ssion chain is provided in Fig. 2.1. The evolution of the ssion chains depends on the balance between the loss and production events, which is quantied using the multiplication factor k = production absorption + leakage .

(2.6)

If k > 1 the system is said to be supercritical and the average number of neutrons will grow in time ; if k < 1 the system is said to be subcritical and the average number of neutrons will decrease in time ; nally, if k = 1 the system is said to be critical and the average number of neutrons will be stationary in time.

. The time-dependent Boltzmann equation

In order to characterize the stochastic evolution of the ssion chains, equations can be established for the statistical moments of neutron and precursor populations. In particular, the conservation law (expressing the net change as the balance between creations and losses) for the neutron density n(r, Ω, E, t),

i.e., the average number of neutrons in an elementary volume in phase space, and the precursor density (2.7)

c j (r, t) for each family j = 1, 2, • • • , M f , M f
The Boltzmann equation, expressed as a function of the angular ux, reads

1 v ∂ ∂t φ(r, Ω, E, t) + Lφ(r, Ω, E, t) = F p φ(r, Ω, E, t) + j χ j d (r, E) 4π λ j c j (r, t) + S(r, Ω, E, t), (2.8)
where we have introduced the net disappearance operator

Lφ = Ω • ∇φ + Σ t (r, E)φ - ν s (r, E ′ )Σ s (r, Ω ′ - → Ω, E ′ - → E)φ(r, Ω ′ , E ′ , t)dE ′ dΩ ′ ,
(2.9)

and the prompt ssion operator

F p φ = χ p (r, E) 4π ν p (r, E ′ )Σ f (r, E ′ )φ(r, Ω ′ , E ′ , t)dE ′ dΩ ′ .
(2.10)

Here Σ t is the total macroscopic cross section, Σ f is the macroscopic ssion cross section, Σ s is the dierential scattering cross section, χ p is the prompt ssion emission energy spectrum, χ j d is the delayed emission energy spectrum for precursor family j, ν p is the average prompt ssion yield, ν j d is the average delayed neutron yield for precursor family j, λ j is the exponential decay constant for family j, and ν s (r, E ′ ) is the particle yield due to scattering. The term S denotes an external source [2].

The net disappearance operator is the sum of the streaming term Ω • ∇φ, expressing the ow out of the elementary volume in the phase space, the collision rate Σ t φ, describing the loss of neutrons by interaction with the medium, minus the scattering term ν s (r,

E ′ )Σ s (r, Ω ′ - → Ω, E ′ - → E)φ(r, Ω ′ , E ′ , t),
describing the rate of transfer of neutrons with energy E ′ and direction Ω ′ to neutrons with energy E and direction Ω. The prompt ssion operator F p φ denotes the production rate of prompt neutrons. Equation (2.8) is coupled to the evolution equations for the precursor densities c j , which read

∂c j (r, t) ∂t = ν j d (r, E ′ )Σ f (r, E ′ )φ(r, Ω ′ , E ′ , t) dE ′ dΩ ′ -λ j c j (r, t), (2.11)
The term ν j d (r, E ′ )Σ f (r, E ′ )φ(r, Ω ′ , E ′ , t) expresses the production rate of precursors due to ssion events. On physical grounds, we can neglect any external precursor source.

The system of Eqs. (2.8) and (2.11) is closed by applying initial and boundary conditions to φ and the c j 's. The solution of Eqs. (2.8) and (2.11) fully characterizes the time evolution of the average neutron density and the average precursor density in the phase space.

. Timescale separation between neutrons and precursors

It is important to note that the typical timescales of neutrons and precursors in nuclear reactors are vastly dierent. The average neutron generation time Λ, i.e., the average time between the birth of a neutron by ssion and its disappearance by the subsequent ssion event, is about Λ ≃ 0.1 µs for fast reactors, and Λ ≃ 20 µs for light-water-moderated reactors. The delayed neutron fraction is β ≃ 0.003 for fast reactors, and β ≃ 0.006 for light-water-moderated reactors. The family-averaged precursor decay time is about λ -1 ≃ 10 s for both systems. Thus, ssion chains at equilibrium (i.e., for a critical reactor) generally take the form of short outbursts of prompt neutrons that die out rapidly, over a few Λ (the system is subcritical without the contribution of the delayed neutrons) ; at each ssion event, a small fraction of the ssion neutrons is produced in the form of precursors, which much later emit additional delayed neutrons that are responsible for the chain to be critical and thus stationary. Due to the values of the physical constants involved, the number of neutrons circulating in the reactor at equilibrium is much smaller than the number of precursors : this can be quantied by the dimensionless parameter θ = (λΛ/β), which expresses the ratio between the population sizes of neutrons and precursors. For fast reactors, θ ∼ 10 -6 , and for light-water-moderated reactors θ ∼ 10 -4 .

The unbalance between the two timescales is actually highly benecial for reactor control : thanks to the presence of the delayed neutrons, the kinetics of the neutron population is slowed down by a factor θ compared to the case where precursors are neglected. However, on the computational side, this also means that the system of Eqs. (2.8) and (2.11) is sti, which poses distinct challenges to both deterministic and Monte Carlo solvers, calling for specic numerical methods and clever modications of the algorithms conceived for stationary problems. In particular, the strategies required for kinetic Monte Carlo simulation will be illustrated in Sec. 2.3.

. Determination of the equilibrium source

Most often, the initial condition for the system of Eqs. (2.8) and (2.11) corresponds to the reactor being critical. The stationary initial state of the neutron and precursor density is then determined by formulating an eigenvalue problem for the Boltzmann equation and solving for the associated fundamental eigenmode.

More precisely, this condition is obtained by setting the time derivatives to zero in Eqs. (2.8) and (2.11), neglecting the external source, and then plugging the resulting stationary Eqs. (2.11) for the precursor population into the stationary Boltzmann Eq. (2.8). This leads to Lφ(r, Ω, E) = Fφ(r, Ω, E), (2.12) where F is the stationary ssion operator dened by

Fφ(r, Ω, E) = F p φ(r, Ω, E) + j χ j d (r, E) 4π ν j d (r, E ′ )Σ f (r, E ′ )φ(r, Ω ′ , E ′ )dE ′ dΩ ′ (2.13)
and φ(r, Ω, E) is the stationary neutron angular ux.

In general, Eq. (2.12) does not necessarily admit a non-null solution φ(r, Ω, E). To ensure the existence of a stationary state, the ssion terms are rescaled by suitable constants k i , yielding an eigenvalue problem for the discrete spectrum of eigenpairs {k i , φ k i } :

Lφ k i (r, Ω, E) = 1 k i Fφ k i (r, Ω, E), (2.14)
where φ k i are the eigenstates associated to the k i -eigenvalues [2]. It is assumed that the eigenvalues k i can be ordered, with

• • • ≤ k i ≤ k i-1 ≤ • • • ≤ k 2 ≤ k 1 < k 0 ,
where k 0 > 0 is the fundamental eigenvalue, with multiplicity one. The question of the completeness of the spectrum of k-eigenfunctions as a basis to expand the stationary angular ux φ(r, Ω, E) is non-trivial and problem-dependent [2]. Among these eigenpairs, the fundamental eigenstate φ k 0 ≥ 0 is associated to the largest eigenvalue k 0 and is the only eigenstate that is everywhere non-negative. It is endowed with special interest as it corresponds to the equilibrium state of the reactor. The parameter R = k 1 /k 0 < 1, i.e. the ratio between the eigenvalue k 1 associated with the rst excited eigenstate and the fundamental eigenvalue k 0 , takes the name of dominance ratio and characterizes the relaxation of higher-order eigenmodes towards the fundamental eigenmode [2]. The fundamental eigenvalue is sometimes noted k e , and physically corresponds to the multiplication factor, i.e. the ratio of the size of two successive ssion generations.

Once the fundamental eigenstate of the angular ux is known, the distribution of neutrons at t = t 0 corresponding to a critical reactor is obtained from [START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF] n eq (r, Ω,

E, t = t 0 ) = φ k 0 (r, Ω, E) v .
(2.15)

Observe that the neutron source includes contributions from all kinds of collisions, and not just ssion events. The initial distribution of precursors can be correspondingly deduced from Eq. 2.11 in the steady state, yielding

c eq j (r, t = t 0 ) = 1 λ j ν j d (r, E ′ )Σ f (r, E ′ )φ k 0 (r, Ω ′ , E ′ )dΩ ′ dE ′ .
(2.16)

Equations (2.15) and (2.16) dene the equilibrium source for kinetic problems.

. A review of Monte Carlo methods for particle-transport problems

In this section, we present an overview of general Monte Carlo methods for particle-transport problems, before addressing the special case of time-dependent transport problems obeying the system of Eqs. (2.8) and (2.11).

Monte Carlo simulations rely on the stochastic sampling of particle histories : for the applications considered here, particles can be either neutrons or precursors. Sampling is based on pseudo-random number generators, which generate sequences of numbers `behaving as they were random' using deterministic algorithms starting from an initial `seed'. By denition, such sequences are not truly random, and after a given period they repeat themselves ; furthermore, the numbers of the series are correlated. While sophisticated statistical tests and theorems from number theory can provide a sound basis for choosing an appropriate generator for Monte Carlo simulations, in practice the goodness of the pseudo-random number generators must be judged by the quality of the obtained results. Modern pseudo-random generators have very large periods (e.g. ∼ 2 128 for the PCG64 random generator [START_REF]Using the PCG C++ implementation[END_REF], or ∼ 2 19937 for the Mersenne Twister [START_REF] Matsumoto | Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator[END_REF]) and suciently weak correlations ; we shall thus assume that the samples of the generators behave as if they were independently and identically distributed.

There exist two main families of Monte Carlo methods : in analog simulations, the rules of the Monte Carlo game follow closely the physical laws of the underlying transport process ; on the contrary, in non-analog simulations the sampling rules are suitably modied in order to achieve a smaller statistical uncertainty on the estimated observables. In the latter case, the average of the observables is preserved by correspondingly modifying the statistical weight of the particles : this ensures that the non-analog game will yield a fair (unbiased) result with respect to the analog game. The choice of the non-analog games should be guided by considerations related to reducing the statistical uncertainty for a given amount of computer cost involved in the sampling procedure : the arsenal of non-analog sampling techniques goes under the name of variance-reduction methods.

The statistical weights encode the fact that, in non-analog games, particles do not contribute equally to the estimation of the statistical moments of the simulated population. To x the ideas, consider a very simple example : a neutron with initial statistical weight of w = 1 undergoes a collision event ; upon collision, the absorption probability is p a and the survival probability p s = 1 -p a . The analog simulation of this process would be to sample ξ ∼ U[0, 1[, i.e. ξ is sampled uniformly in [0, 1[, and to kill the neutron if ξ < p a : either the neutron survives the collision event, or it dies ; with this algorithm, the statistical weight of the neutron is preserved. An equivalent non-analog scheme is to enforce the survival of the neutron by modifying the statistical weight w of the neutron exiting the collision ; the neutron always survives and its statistical weight is multiplied by the survival probability p s at each collision. In both situations, the average statistical weight of the surviving neutrons after k collisions is E[w] = p k s . The variance of the number of surviving neutrons after k collisions in the analog game will be Var[w] = kp s (1 -p s ), while it will be 0 for the non-analog game.

Often, variance-reduction methods are used in combination with population-control methods, whose aim is to prevent the population (or the statistical weights) from shrinking or exploding. These issues will be discussed extensively in the following.

. Tallying in Monte Carlo simulations

The ultimate goal of Monte Carlo simulations is to provide an unbiased estimate of the average value of the observables of interest, such as the ux, reaction rates, particle current, etc. The Monte Carlo estimation of the sought physical observable is called a score or tally, which is in general a functional of the random walks of the sampled particle histories. Since the evolution of the particle histories is a stochastic process, the tallies are random variables, with unknown distributions. For each history, each sampled event in the Monte Carlo simulation has a (possibly zero) contribution to the score, and the average is computed by taking the sample mean over all the related contributions brought by all particle histories. Contributions to the Monte Carlo score are gathered using estimators. To x the idea, let be the stationary scalar neutron ux integrated over a volume V and over the energy interval ∆E = [E m , E M ]. It is assumed that the material properties are constant in the region V . It can be shown that the corresponding Monte Carlo score can be estimated using the collision estimator

φ V,∆E = V E M Em φ(r, Ω, E)dΩdEdr (2.
h(r, E, w) = w Σ V t (E) , (2.18)
with h(r, E, w) = 0 for r / ∈ V or E / ∈ ∆E. Here w is the statistical weight of the incident neutron, Σ V t (E) is the total cross section value within the region V , and E is the energy of the incident neutron. The estimator h is a deterministic function evaluated at the random collision points with the random coordinates of the incident neutron, and is non-vanishing only when the collision coordinates fall within the integration region V and ∆E. For each neutron, upon collision h is added to the Monte Carlo score, and the mean value of the score is computed over a large ensemble of Monte Carlo histories :

µ[h(r, E)] = i j h(r i,j , E i,j , w i,j ), (2.19)
where i is the index over histories, and j the index over collisions for a given history.

Each simulated history yields an estimate of the sought observables, and is called a replica. Condence intervals for the mean value of the scores can be obtained relying on the Central Limit Theorem by estimating the variance of the scores. Ideally, samples are assumed to be statistically independent, in which case estimating the variance is straightforward. In literature, the variance associated to the ensemble average denes the true variance [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF]. In some situations (as discussed in Sec. 1.2.1), independence is not veried and more complex inference techniques are necessary. A thorough and rigorous introduction to estimators and tallies can be found in Ref. 10.

. Example of a typical analog Monte Carlo simulation

In the simplest analog framework, neutron histories are simulated by closely following the elementary physical laws that describe neutron-matter interactions, which can be retrieved from the nuclear data libraries. For instance, a neutron moving in a homogeneous medium would sample ight lengths from an exponential distribution of parameter Σ t (E), once the emission direction Ω and energy E are known :

f (s) = Σ t (E)e -Σ t (E)s .
(2.20)

Using the inverse transform method, the ight length can be sampled from

s = - 1 Σ t (E) ln(1 -ξ), (2.21)
where ξ ∼ U[0, 1[.

If the spatial position of the particle after the ight falls within the system (i.e., if the particle is not lost upon leaving the boundaries), the particle will undergo a collision event.

p i,k = σ i,k (E) σ i,t (E) , (2.22)
with

k p i,k = 1.
If the event is scattering or ssion, the number and features of the secondary particles are sampled according respectively to the yield and energy-angle distribution of the chosen reaction channel. For instance, for isotropic elastic scattering (with unit yield ν s = 1) one would rst sample the scattering cosine µ C in the center-of-mass frame, uniformly in the interval -1 ≤ µ C ≤ 1, and the outgoing energy E ′ would be determined by the collision kinematics as

E ′ = E A 2 + 2µ C A + 1 (A + 1) 2 ≤ E, (2.23)
where A is the ratio of the mass of the collided nuclide with respect to the mass of the neutron, and E is the energy of the incident neutron. Since µ C is random, E ′ is also random, and is correlated to µ C . The azimuthal angle is taken to be uniformly distributed in [0, 2π]. For ssion, the energy of the secondary neutrons would be sampled from the ssion spectra χ p (E) and χ j d (E), which are in most cases only weakly dependent on the energy of the incident neutron. In order to sample the number n f of secondary neutrons, one would need the full distribution p(n f ) of the ssion yield : in practice, it turns out that in most nuclear data libraries the full distribution is not provided, and the only available piece of information is the average prompt and delayed neutron yield. Thus, a full analog sampling of the number n f of secondary neutrons is not possible, and for historical reasons the most commonly adopted choice for Monte Carlo neutron-transport codes is to sample the integer numbers surrounding the value of the average yield ; the probability of drawing either one is determined so that the average number of sampled neutrons is precisely equal to the average ssion yield. For the prompt neutrons, for instance, we have

n f = ⌊ξ + ν p ⌋ , (2.

24)

with ξ ∼ U[0, 1[. Note that, for the purpose of solving for the average neutron density or neutron ux, as described by the Boltzmann equation, only the average ssion yields are required, and the sampling procedure for the multiplicity only aects the variance of the tallies.

. Overview of basic non-analog methods

The simple example discussed in Sec. 2.2 shows that introducing non-analog sampling strategies can be eective in reducing the variance associated to the scores. In this section, we provide a brief overview of the most common variance-reduction and population-control techniques that will be used throughout this work. For the sake of conciseness, we only sketch their basic features, and implementation details are left for discussion when they are used in specic applications.

Implicit capture and forced ssion

Implicit capture is perhaps one of the best-known non-analog collision sampling strategies. First, we assume a non-ssile medium, so that neutrons undergo either sterile absorption or scattering. To prevent neutron histories from being terminated by sterile capture without contributing to scores, one can resort to implicit capture : upon collision, the neutron is forced to survive (i.e., to undergo scattering), and its statistical weight is multiplied by the survival probability

w ′ = w Σ s Σ t , (2.25)
where w ′ is the weight of the outgoing neutron and w is the weight of the incident neutron. The ratio of scattering to total cross section is evaluated at the incident neutron coordinates. With implicit capture, neutron histories can only be terminated by leakage, which suggests that the use of Russian roulette (see Sec. 2.2.3) can be benecial to remove particles whose statistical weight has become insignicant for the simulation.

When ssion is taken into account, implicit capture can be combined with forced ssion, whose aim is to promote the birth of secondary neutrons and thus the sampling of additional ssion chains.

In this case, at each collision site the incident neutron is forced to undergo scattering, as above. The statistical weight of the emitted neutron is corrected as in Eq. (2.25). Furthermore, provided that the collided nuclide is ssile, a ssion event is systematically sampled, and the prompt and delayed ssion yields are adjusted as ν p Σ f /Σ t or ν d Σ f /Σ t , respectively, in order to ensure a fair game. This means that n p,f prompt ssion neutrons are sampled following

n p,f = ξ + ν p Σ f Σ t , (2.26)
where ξ ∼ U[0, 1[, and are assigned the statistical weight w of their parent neutron. Furthermore, ν d,f delayed neutrons (or, in the case of kinetic simulations, precursors : see Sec. 2.3) are also sampled following

n d,f = ξ + ν d Σ f Σ t , (2.27)
and are assigned the statistical weight of their parent neutron. This is the technique used e.g. in the Monte Carlo code TRIPOLI-4 ® , developed at CEA [START_REF] Brun | TRIPOLI-4®, CEA, EDF and AREVA reference Monte Carlo code[END_REF]. Other implementations of the forced ssion algorithm exist [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF][START_REF]X-5 Monte Carlo Team[END_REF], but they will not be discussed here.

Branchless collisions

Branchless collisions have been popularized by Lux and Koblinger [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF] and applied with success rst to xed source stationary simulations in Ref. [START_REF] Sjenitzer | Variance reduction for xed-source Monte Carlo calculations in multiplying systems by improving chain-length statistics[END_REF], and then to kinetic simulations [START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF][START_REF] Molnar | A GPU-based direct Monte Carlo simulation of time dependence in nuclear reactors[END_REF]. The main idea of branchless collisions is that each collision leads to a single outgoing particle that is assigned the same statistical weight multiplier, regardless of the chosen reaction channel :

w ′ = w ν k Σ k Σ t , (2.28)
where ν k is the yield associated to reaction channel k. For instance, if k is the scattering reaction as dened in Sec. 2.1.1, its yield is ν s . If it is the prompt ssion channel, then ν k = ν p , and if it is the delayed ssion channel, then ν k = ν d , and so on. The reaction channel k is sampled with probability

P k = ν k Σ k ν k Σ k , (2.29)
and the corresponding interaction leads to the production of exactly one particle for the sampled channel.

The features (direction, energy, etc.) of the outgoing particle are sampled as in Sec. 2.2.2. Note that, in time-dependent Monte Carlo simulations, choosing the delayed ssion channel leads to the `replacement' of the incoming neutron by a precursor particle, as discussed later. Branchless collisions are actually a generalization of implicit capture, and reduce to implicit capture in non-multiplying media. Note in particular that neutron histories can only be terminated by leakage, encouraging the use of additional population-control algorithms.

Russian roulette and splitting

When particles are assigned statistical weights, it is legitimate to consider the trade-o between increased calculation time and the statistical signicance of simulated particles. It may be then advisable to control the distribution of the statistical weights to avoid needlessly spending calculation time on particles that do no matter much for the estimation of scores. Such `population control' methods must be unbiased in order to preserve the average scores.

Russian roulette is one the simplest ways to ensure that the statistical weight do not become too low [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. When the statistical weight w of a particle falls below a threshold w R , the particle plays Russian roulette ; its weight is updated in the following way :

w ′ = w surv with probability w/w surv w ′ = 0 otherwise, (2.30)
i.e. the particle is either assigned the survival weight w surv > w R , or killed (and thus removed from the Monte Carlo simulation). Note that the particle weight is conserved on average.

Another common variance reduction technique is splitting. When the statistical weight w of the particle becomes higher than a threshold w S , the particle is split in n s = ⌊w/w S ⌋ identical copies, and each copy is assigned a statistical weight w ′ = w/n s . Again, the particle weight is conserved on average.

One should note that these two algorithms can be applied at any stage of the simulation ; in particular, they can be applied after collisions or before collisions. The eects of the combination of Russian roulette and splitting is illustrated in Fig. 2.2.

Weight combing

Assume a population of N particles of total statistical weight W = n w n , where the w n might have a large statistical dispersion. We want to comb this population into a new population of M particles sharing the same weight w av = W/M and having the same total weight M × (W/M ) = W as the initial population. Furthermore, we require combing to be unbiased, in the sense that the average weight of individual particles should be preserved [START_REF] Booth | A weight (charge) conserving importance-weighted comb for Monte Carlo[END_REF].

The combing procedure is better explained by a scheme, and one can refer to Fig. 2.3 for illustration.

Essentially, each `tooth' of the comb corresponds to a copy of a particle in the initial population : if the weight of a particle is large and several teeth fall on the same particle, it will be copied several times and each copy will be assigned the statistical weight w av . Conversely, if the weight of a particle is small, then so is the probability that a tooth falls on it and the particle might not be copied at all. It can be shown that the unbiasedness of the combing scheme is ensured by selecting the rst tooth randomly, at [START_REF] Sjenitzer | A Monte Carlo method for calculation of the dynamic behaviour of nuclear reactors[END_REF][START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF][START_REF] Hoogenboom | Extensions of the MCNP5 and TRIPOLI4 Monte Carlo Codes for Transient Reactor Analysis[END_REF]. These authors have paved the way towards an ecient implementation of kinetic Monte Carlo schemes fully taking into account precursors. Although their results stem from a great amount of ingenuity in conceiving and rening algorithms, it must be also acknowledged that these methodological advances have greatly beneted from the considerable increase in available computer power over the last ten years.

position ξ × W/M ,
One of the key point of the scheme proposed by Sjenitzer and Hoogenboom is the use of ad hoc variance-reduction methods targeting the time variable. The introduction of a collapsed precursor particle, coupled with a forced decay scheme, allows considerably reducing the statistical dispersion due to the timescale of precursors, while preserving the unbiasedness of the scores [START_REF] Legrady | Scouting the feasibility of Monte Carlo reactor dynamics simulations[END_REF]. The introduction of branchless collisions, including a special modication to account for precursors, allows mitigating the statistical dispersion due to the appearance of branches in neutron histories at each ssion event [START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF].

The introduction of population-control mechanisms such as Russian roulette, splitting and weight combing ensures that the population size does not diverge nor vanish in the course of the kinetic simulation [START_REF] Sjenitzer | A Monte Carlo method for calculation of the dynamic behaviour of nuclear reactors[END_REF][START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF].

The strategy of Sjenitzer and Hoogenboom has been implemented over the last few years in several production-level Monte Carlo codes, sometimes with algorithmic improvements, variants and modications [START_REF] Levinsky | Modeling of the SPERT transients using Serpent 2 with time-dependent capabilities[END_REF]6062]. Intense research eorts have been devoted to kinetic Monte Carlo methods, as witnessed by the European H2020 McSAFE project (2017-2020), which was devoted to the industrialization of such methods in view of enhancing their reliability and ease of use for real-world applications [START_REF] Demazière | Advanced numerical simulation and modelling for reactor safety -contributions from the CORTEX, HPMC, McSAFE and NURESAFE projects[END_REF].

Within this framework, kinetic Monte Carlo simulations have been validated against the experimental data of the reactivity insertion accident (RIA) test campaign carried out in the 1960s on the SPERT III E-core [START_REF] Mancusi | Monte Carlo simulations of the SPERT III E-core transient experiments[END_REF][START_REF] Ferraro | Serpent/SUBCHANFLOW pin-by-pin coupled transient calculations for a PWR minicore[END_REF]. Extensive verication based on code-to-code comparisons on reactor benchmarks has allowed singling out possible performance issues and identifying future research paths concerning algorithmic and implementation improvements [START_REF] Faucher | Variance-reduction methods for Monte Carlo kinetic simulations[END_REF][START_REF] Ferraro | Serpent and TRIPOLI-4 transient calculations comparisons for several reactivity insertion scenarios in a 3D PWR minicore benchmark[END_REF]. Among current research trends concerning kinetic Monte Carlo methods, a special role is played by GUARDYAN, the only production-level Monte Carlo code designed specically for time-dependent Monte Carlo simulations and entirely based on GPUs [START_REF] Molnar | A GPU-based direct Monte Carlo simulation of time dependence in nuclear reactors[END_REF].

The GPU architecture implies massive changes in the way Monte Carlo simulation is performed, but has shown potential for excellent performances [START_REF] Legrady | Full core pin-level VVER-440 simulation of a rod drop experiment with the GPU-based Monte Carlo code GUARDYAN[END_REF].

In the next section, we review the most frequently used Monte Carlo methods for kinetic Monte Carlo simulations, including the sampling of the equilibrium source distribution. The general concepts are recalled here, whereas the details of the more complex algorithms are introduced in Chapters 5-6, where they are specically needed. The overall simulation scheme is illustrated in Algorithm 1 and will (2.14), which can be estimated by Monte Carlo methods using the well-known power iteration method [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. Starting from an arbitrary tentative ssion source Q, a `neutron generation' is sampled : each history of the initial particle population consists of a succession of ights and collisions, whereupon neutrons undergo scattering, ssion, or capture.

The neutrons are followed until they are absorbed (by ssion or capture) or leak out of the system.

Upon ssion, on average, ν descendant neutrons are sampled and stored in a `ssion bank' (using for instance forced ssion), and are not further followed during the current generation. When all the histories of a given generation have been sampled, the neutrons in the ssion bank are tested (if required)

for population control, and neutrons surviving population control are promoted to source neutrons for the next generation. In addition, before starting the next generation, the weight of each neutron is divided by the generation-wise estimate of the multiplication factor, so that all generations start with the same total weight. It can be shown that, after a suciently large number of generations, the ssion source distribution reaches a statistical equilibrium, and subsequent generations will be identically but not independently distributed. The lack of independence is due to the fact that successive generations are by construction correlated, the ssion source for a given generation stemming from the ssion sites of the previous generation. The ssion source converges towards its fundamental eigenmode, and the ratio between the statistical weights of two successive ssion sources converges towards the fundamental eigenvalue k 0 . The rate of convergence of the power iteration from an arbitrary initial condition to the stationary state is primarily driven by the dominance ratio R, i.e. the ratio k 1 /k 0 [START_REF] Ueki | Autocorrelation and dominance ratio in Monte Carlo criticality calculations[END_REF]. The generations that are needed to reach equilibrium are usually called inactive, and are discarded for the purpose of collecting tally statistics, since during this phase of the power iteration the particle distribution has not yet reached its asymptotic stationary behaviour.

For historical reasons, mainly motivated by considerations of computation cost, the sought averages are estimated by running the power iteration for a sucient number of generations after convergence has been reached, and collecting the required statistics over these so-called active generations. Thus, the approach used by Monte Carlo practitioners for power iteration is basically an ergodic average, in analogy with equilibrium statistical mechanics. This is to be compared with the standard approach that one would apply to estimate ensemble averages over independent replicas, in which case a collection of power iteration simulations would be needed, each starting from a tentative ssion source and having a sucient number of inactive generations until convergence, plus a nal generation to collect tallies.

The ergodic approach to power iteration has been preferred because the involved computer cost is much lower, although this comes at the expense of introducing correlations in the estimated observables. In the literature, the variance obtained using ergodic averages is dubbed `apparent', in contrast with the true variance that would be obtained running independent replicas [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF]. The ergodic and ensemble average approaches for power iteration are illustrated Fig. 2.4 and 2.5, respectively.

. Sampling the stationary source for kinetic problems

Let us assume that the neutron population is distributed according to the fundamental eigenmode φ k 0 (r, Ω, E) : we have recalled in Sec. 2.2.1 that an unbiased estimator for the angular ux consists in tallying the quantity w/Σ t (r, E) at each collision. Based on Eq. (2.15), the estimator for the initial neutron density at position r and energy E is thus r,E) .

h n = w vΣ t (
(2.31)

The precursor density for a family j can be simultaneously sampled, provided that the current material is ssile, using the estimator ,E) .

h j c = w ν j d λ j Σ f (r, E) Σ t (r
The statistical dispersion due to the large variation of the decay constants λ j can be considerably reduced by introducing the combined (or collapsed) precursor particle, as dened by Sjenitzer and Hoogenboom [START_REF] Sjenitzer | A Monte Carlo method for calculation of the dynamic behaviour of nuclear reactors[END_REF], a representative particle whose statistical weight depends on the decay rate for each family. The combined precursor density at equilibrium can be estimated using

h c = wν d λ Σ f (r, E) Σ t (r, E) , (2.32)
where λ =

β j β j λ j
is the family-averaged decay rate of the combined precursor. It is important to stress that the collapsed precursor sampling scheme is unbiased.

Because the number of collisions, their positions, etc. are random variables, the total statistical weight of the kinetic source is a random variable, which has to be normalized to a known value at the start of the kinetic simulation. It is customary to start the time-dependent simulations with the same total weight as in the calculation leading to the estimation of the fundamental eigenmode φ k 0 (r, Ω, E) (see Sec. 2.4), although it is possible to choose a dierent value. In addition, population control can be applied on the kinetic source.

. Sampling the time-dependent process

Let us assume now that the neutron and precursor source needed for time-dependent transport has been sampled, e.g. following the procedure in Sec. 2.4.1. Starting from the source, particle histories are followed through time as described in Algorithm 1. Particles beginning their life at t = t 0 are assigned a time label, in addition to the space, direction and energy coordinates that are required for stationary problems.

The tracking of particles over time [t 0 , T ] is decomposed in discrete (and not necessarily evenly spaced) time intervals [t i , t i+1 ] : particles are simulated until the end of each time-step, whereupon a population control algorithm such as combing, Russian roulette, and splitting is possibly applied to neutrons and precursors, in order to control the dispersion of statistical weights as well as to keep the total number of simulated particles within acceptable limits. Since neutron and precursor transport is a Markov process, particles can be stopped at arbitrary times without incurring any bias. Thus, the time mesh for kinetic Monte Carlo simulations has to be contrasted with the time mesh used in deterministic calculations, which eectively induces discretization errors in the obtained results.

In addition, a time-grid has to be dened to allow scoring time-dependent tallies. This time-grid does not need to coincide with the time-grid for population control, although it is often the case. Some tallies are based on snapshots of the neutron and/or precursor population at the end of the time bin ; however, most tallies, such as neutron ux or reaction rates, are integrated over each time-window, which means that all contributions falling within the current time bin are summed in the estimator. This procedure is again unbiased.

The life of a neutron during a time-step is simulated as described in Sec. 2.2.2. At each travelled ight length l at speed v is associated a displacement in time by the amount δt = l/v. The neutron is followed until it undergoes a termination event (leakage, capture in the case of analog simulations, Russian roulette, etc.) or it reaches the end of the current time-step. If the ight duration takes the neutron after the end of the time-step, the neutron is transported until the end of the time-step, and then added to the bank of neutrons that collectively undergo population control before beginning the following time-step. Secondary neutrons and precursors originating from ssion events within a time step are stored in a bank of secondary particles whose histories are simulated once the history of the primary particle has been terminated. Due to ssion, the typical history will have `branches', which generally increase both the statistical dispersion and the computer time. Observe in particular that the presence of branches induces non-trivial correlations between particles being present at distinct sites and distinct times, since they might stem from a common ancestor. To cope with the increased variance and computational cost inherent to kinetic simulations, the collision sampling strategy is generally non-analog and encompasses techniques such as branchless collisions (see Sec. 2.2.3).

In an analog decay strategy, precursors are sampled at ssion events and emit delayed neutrons following their exponential decay law with parameter λ j . The analog emission time of the delayed neutron is sampled as

τ = t b - 1 λ j ln(1 -ξ), (2.33)
where t b is the birth time of the precursor and ξ ∼ U[0, 1[. Upon emission, the delayed neutron behaves as any other neutron would.

However, due to the large separation between the timescale of neutrons and the one of precursors, as mentioned earlier, the number of neutrons at a given time is much smaller than the number of precursors when using analog sampling, which motivated the introduction of the `forced precursor decay' strategy [START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF][START_REF] Legrady | Scouting the feasibility of Monte Carlo reactor dynamics simulations[END_REF]. The idea is to introduce a third time-grid for the simulation, which generally coincides with the tallying time-grid, and enforce precursor decay in each time-step : a fraction of the precursor's statistical weight always survives, related to the probability that the precursor does not decay during the time-step. The forced decay time is generally chosen uniformly in each time-step, and precursors are not eliminated upon decay.

After the decay time is sampled, the statistical weight for precursors and for the subsequent delayed neutrons is determined as follows. For a precursor originating from the equilibrium source distribution, the statistical weight reads

W eq c (t|t 0 ) = w c j λ λ j β j β e -λ j (t-t 0 ) , (2.34)
where w c it the precursor weight at production time and t 0 is the initial time of the equilibrium source.

The remaining fraction of statistical weight leads to the production of a delayed neutron at time t with adjusted statistical weight. Correspondingly, the statistical weight of the delayed neutron is

W eq delayed (t|t 0 ) = w c (t i+1 -t i ) j λ β j β e -λ j (t-t 0 ) , (2.35)
where w c is the parent precursor weight at the production time of the parent precursor.

The statistical weight for a precursor produced in the course of the kinetic simulation reads

W c (t|t 0 ) = w c j β j β e -λ j (t-t 0 ) , (2.36)
where here t 0 denotes the production time (i.e., the time of the ssion event) and w c the associated statistical weight. The corresponding statistical weight of the delayed neutron reads

W delayed (t|t 0 ) = w c (t i+1 -t i ) j λ j β j β e -λ j (t-t 0 ) .
(2.37)

Upon decay, the actual family of the combined precursor has to be sampled in order to determine the energy of the delayed neutron. The probability for the delayed neutron to originate from family j is given by

p(j, t|t 0 ) = λ j β j β e -λ j (t-t 0 ) k λ k β k β e -λ k (t-t 0 )
.

(2.38)

When using this sampling strategy, the population size of precursors can only increase : precursors are not eliminated by decay anymore, and their number increases due to ssion. This suggests the use of population control algorithms to keep the population size manageable. For a thorough discussion, see e.g. Ref. 62.

. Simplied formulations of the transport equations

The system of Eqs. (2.8) and (2.11) provides a complete description of reactor kinetics. For some applications, if one is interested in trading accuracy for computer time, these equations can be simplied introducing suitable hypotheses. The main interest of such strategy is for deterministic solvers, which in any case must rely on a discretized, and thus approximated, version of the Boltzmann and precursor equations. Nonetheless, simplied formulations of Eqs. (2.8) and (2.11) may also be interesting to obtain closed-form solutions to be compared with Monte Carlo simulation results, or to considerably reduce the computation cost of Monte Carlo simulations. In the following we will briey examine some of these approaches, detail the resulting equations, and for each propose a Monte Carlo counterpart.

. The multi-group approximation

The multi-group formulation of the transport equation consists in integrating the continuous energy variable E over a nite number G of energy intervals [E g , E g-1 ], g = 1, • • • , G, which are called groups.

Correspondingly, the multi-group angular ux is dened by

φ g (r, Ω, t) = E g-1 Eg φ(r, Ω, E, t)dE.
(2.39)

Integrating Eqs. (2.8) and (2.11) over the energy intervals [E g , E g-1 ] yields the system of discretized equations

1 v g ∂ ∂t φ g + L G φ g = F G p φ g + j χ j d,g 4π λ j c j (r, t) + S g , (2.40)
and

∂c j (r, t) ∂t = g ′ ν j d,g ′ Σ f,g φ g ′ (r, Ω ′ , t) dΩ ′ -λ j c j (r, t).
(2.41)

The stationary formulation can be expressed similarly. In order to establish Eqs. (2.40) and (2.41), We have made use of the following denitions : the multi-group net loss operator reads

L G φ g = Ω • ∇φ g + Σ t,g φ g - g ′ ν s,g ′ Σ s,g ′ →g (r, Ω ′ → Ω)φ g ′ (r, Ω ′ , t)dΩ ′ , (2.42)
and the multi-group prompt ssion operator reads

F G p φ g = χ p,g 4π g ′ ν p,g ′ (r)Σ f,g ′ φ(r, Ω ′ , t)dΩ ′ . (2.43)
By inspection, it is apparent that the multi-group constants (cross sections, spectra, etc.) that occur in Eqs. (2.40) and (2.41) must be of the form

Σ k,g (r, Ω) = E g-1 Eg Σ k (r, E)φ(r, Ω, E, t)dE φ g (r, Ω, t) (2.44)
for the cross sections of reaction k, and

Σ s,g ′ →g (r, Ω) = E g-1 Eg E g ′ -1 E g ′ Σ s (r, Ω ′ → Ω, E ′ → E)φ(r, Ω, E, t)dE dE ′ φ g ′ (r, Ω ′ , t) , (2.45)
for the scattering dierential cross section. The other constants of the multi-group equations have similar angular-ux-weighted expressions.

Observe that the weighting factor appearing in the multi-group constants is precisely the angular ux φ(r, Ω, E, t), which is the unknown solution of the original system of Eqs. (2.8) and (2.11). Therefore, in order to achieve a rigorous derivation of the parameters of the multi-group equations one should have already solved the continuous-energy problem, which is what we wanted to avoid in the rst place.

Furthermore, observe that now the multi-group cross sections depend also on the particle direction Ω, contrary to the regular continuous-energy cross section, due to the dependence on φ(r, Ω, E, t). The problem of computing reliable multi-group constants has attracted enormous research eorts, since the beginning of the era of computational methods for reactor physics problems, and goes well beyond the scope of this manuscript. In the context of the numerical methods developed for deterministic solvers, it is common practice to inject into the denitions of the multi-group constants a tentative ux shape, derived by carefully taking into account self-shielding eects using state-of-the-art models (see e.g. [65 69]). More recently, Monte Carlo codes have been successfully used in order to estimate the multi-group constants [7073].

The analysis of uctuations and correlations in kinetic Monte Carlo problems carried out in Chapter 6 will be based on Eqs. (2.40) and (2.41), in order to keep the computation cost within reasonable limits.

For the benchmark calculations performed in this context, the multi-group constants are given in the literature. This means that the Monte Carlo sampling procedures outlined in Sec. 2.2 and in Sec. 2.3 for continuous-energy particle transport can be immediately transposed to the multi-group case. Each particle is assigned an index g for the group, instead of the continuous energy variable E. The particle speed is also replaced by the group speed v g . Cross sections are dealt with exactly as in the continuous-energy case ; the scattering transfer kernel is now a matrix, and the outgoing group g ′ can be straightforwardly sampled (conditioned to the incident energy group g) once the matrix elements have been properly normalized. Position and direction are not discretized.

. The diusion approximation

In Chapter 3-4, we will derive analytical results for the statistical moments of the uctuations and correlations in kinetic Monte Carlo methods, which will pave the way towards a sound and reliable framework for the interpretation of the simulation results. For this purpose, we will need closed-form expressions for the rst few moments (and other related observables) of the neutron and precursor populations. The multi-group Eqs. (2.40) and (2.41) are in general not amenable to exact solutions, which is why we will further simplify the underlying equations by introducing the diusion approximation.

There exist several ways of introducing the diusion equation based on the Boltzmann equation [2].

In one approach, the derivation of the diusion equation relies on the so-called P N expansion : the Boltzmann equation is integrated over the direction Ω and one tries to single out terms depending on the scalar ux

φ(r, E, t) = φ(r, Ω, E, t)dΩ (2.46)
alone. Usually, the derivation is done by neglecting energy, and then rephrasing the resulting equation into a multi-group formalism. The key idea is that the equation for φ(r, t) obtained integrating the Boltzmann equation over the direction is not closed, since it depends on the current J(r, t) = Ωφ(r, Ω, t)dΩ.

(2.47)

The P 0 equation reads

1 v ∂φ ∂t + ∇ • J + Σ a (r)φ = ν p Σ f (r)φ + j λ j c j (r) + Q (2.48)
and is basically a conservation law, with Σ a (r) = Σ t (r) -Σ s (r) and Q the angle-integrated external source. The equations for the precursor concentrations c j (r, t) follow from

∂c j ∂t = ν j d Σ f (r)φ -λ j c j (r).
(2.49)

Observe that no approximations have been introduced so far : the P 0 Eqs. (2.48) and (2.49) are exact (in a single-speed framework), albeit not closed, since Eq. (2.48) depends on the unknown quantity J(r, t), whose equation must be determined.

The P 1 equation for J(r, t) can be then written down multiplying the Boltzmann equation by Ω and integrating again over directions, in order to single out terms depending on J(r, t) and on φ(r, t) alone. It turns out that the equation for J(r, t) is not closed either, since it involves higher-order moments. We obtain a hierarchy of P N equations, each being exact (up to the fact that we have neglected the energy dependence) but not solvable, as the hierarchy is innite.

The diusion approximation is obtained by truncating the hierarchy at the order P 1 , assuming that the current and the scalar ux are related by

J(r, t) = -D(r)∇φ(r, t), (2.50)
which takes the name of Fick's law, by analogy with the similar relation occurring in the equation describing the transport of mass. In Eq. (2.50), the quantity

D(r) = 1 3Σ tr (r) (2.51)
is the diusion coecient, with Σ tr (r) = Σ t (r) -μΣ s (r) the so-called transport cross section, and μ the average scattering angle. For isotropic scattering, for example, μ = 0. Injecting Fick's law into the P 0 equation, we nally obtain the diusion equation for single-speed particles, namely,

1 v ∂φ ∂t -∇ • (D(r)∇φ) + Σ a (r)φ = ν p Σ f (r)φ + j λ j c j (r) + Q(r), (2.52)
to be coupled with the precursor Eq. (2.49). We have assumed that the source is isotropic.

For the special case where the material properties are spatially homogeneous, we have ∇•D∇ = D∇ 2 , and the diusion equation takes the form

1 v ∂φ ∂t -D∇ 2 φ + Σ a (r)φ = ν p Σ f (r)φ + j λ j c j (r) + Q(r), (2.53)
Once the single-speed diusion equation has been introduced, a multi-group version can be obtained based on the analogy with the multi-group formalism for the Boltzmann equation, although a rigorous derivation is much more involved than in the single-speed case. This would read

1 v g ∂φ g ∂t -∇ • (D g ∇φ g ) + Σ t,g φ g = g ′ Σ s,g ′ →g φ g ′ + χ p,g g ′ ν p,g ′ Σ f,g ′ φ g ′ + j λ j c j + Q g , (2.54)
coupled with

∂c j ∂t = χ j d,g g ′ ν j d,g ′ Σ f,g ′ φ g ′ -λ j c j .
(2.55)

The diusion equation can be solved using Monte Carlo simulation (assuming again that all the constants occurring in the equation are known). However, contrary to the multi-group version of the Boltzmann equation, the diusion equation calls for an entirely dierent distribution for the particle displacements. The diusion equation for neutrons has the same functional form as the well-known diusion equation related to e.g. the heat equation, for which a rigorous mapping based on the Feynman-Kac formalism allows phrasing the solution of the equation in terms of a Wiener stochastic process, better known as Brownian motion [START_REF] Kac | On some connections between probability theory and dierential and integral equations[END_REF]. For the case of the neutron diusion equation, the ssion term (which has no analogue in the heat equation) can be dealt with by endowing the Brownian motion with the possibility of branching [START_REF] Kesten | Branching brownian motion with absorption[END_REF].

For the sake of conciseness, we will detail here the simpler case of the single-speed diusion equation in homogeneous materials, with a single precursor family. Suppose that, at a given time, N (t) neutrons and M (t) precursors are present in the system. Neutrons undergo regular Brownian motion with diusion coecient D, until an event occurs. For each neutron, the ssion rate is vΣ f , and the capture rate is vΣ c : the distribution of these events being exponential in time, the time to the rst neutron interaction is sampled as

δt i = - 1 v(Σ c + Σ f )N (t) log(1 -ξ), (2.56)
with ξ ∼ U[0, 1[. For precursors, the only event is decay, which happens at rate λ : the time to the rst precursor decay is sampled as

δt d = - 1 λM (t) log(1 -ξ), (2.57)
with ξ ∼ U[0, 1[. We take then the time increment to the next event as δt = min(δt d , δt i ). The motion

of individual neutrons is sampled from δx ∼ N (0, 2d D δt), (2.58)
where N (µ, σ 2 ) is the normal law of mean µ and variance σ 2 , and d is the dimension. If required, boundary conditions are applied. When all new neutron positions have been sampled, the stopping event is treated : upon ssion, ν p prompt neutrons and ν d precursors are produced on average, as customary ; upon capture, the neutron is removed from the simulation ; and upon decay a precursor is transformed into a delayed neutron. If δt is larger than the duration of the time interval that is used for tallying the physical observables of interest, the population is stopped at the end of the time step and then restarted without memory, which does not introduce any bias because of the Markov nature of the underlying stochastic process.

. Beyond the average : uctuations and correlations

Throughout this Chapter, we have been concerned with establishing the neutron and precursor transport equations in the kinetic regime, either in their exact formulation or under some simplifying hypotheses, and we have shown how these equations can be solved by Monte Carlo methods sampling the random walks of neutrons and precursors in the phase space.

Thanks to the increasing computer power, it becomes possible today to address non-stationary problems by Monte Carlo simulation, which allows for an `exact' treatment of the geometry and of the physics of the simulated system, introducing virtually no approximations nor discretization eects. Monte

Carlo simulations therefore produce unbiased estimates of the rst moments of the sought observables, which are precisely the solutions of the kinetic equations.

However, as detailed in the previous sections and in Chapter 1, such estimates come with a statistical uncertainty that is (asymptotically) inversely proportional to the square root of the number of sampled histories. The almost complete absence of approximations is thus thwarted by the very high calculation cost required to achieve the target accuracy, especially in kinetic simulations. Furthermore, the behaviour of the statistical uncertainty is made more complex by the fact that the simulated particle histories are correlated, the correlations stemming ultimately from ssion events.

Since in most cases Monte Carlo simulations involve some forms of variance-reduction and populationcontrol techniques, the behaviour of uctuations and correlations in kinetic Monte Carlo simulations depends on the non-trivial interplay between the way the sampling strategies are conceived, and the eects inherent to the physics-inspired underlying stochastic processes. In the following Chapters, we will attempt at establishing a rigorous framework that enables to quantify such uctuations and correlations, and thus enhance the reliability of the uncertainty estimation in kinetic Monte Carlo simulation.

. Simulation tools developed for this work

The investigation of correlations occurring in Monte Carlo methods, either in stationary problems over generations or in time-dependent problems over time, could hardly be carried out entirely based on a theoretical framework, and the support of simulations is of course compulsory, especially in the case of complex systems including energy eects and spatial heterogeneity. On the other hand, it turns out that the use of production-level Monte Carlo codes is not an appropriate choice, since correlation functions and related quantities are typically very expensive to compute ; the memory footprint is especially large. On top of that, for our investigation we often need to extensively modify the traditional sampling strategies and implement unusual scores (as discussed in the following chapters), which is in most cases cumbersome due to the complexity of production-level codes.

For these reasons, we have decided to develop our own simulations codes from scratch.

First, we focused on correlations in time-dependent branching Brownian motion, a convenient stochastic representation for diusion-based models. For this purpose, we developed JOFFREY, a Monte Carlo mini-app performing analog simulations of time-dependent branching Brownian motion of a population of neutrons and precursors in a one-dimensional homogeneous, multiplicative medium.

Next, as we focused on more realistic systems, we implemented COYOTE, a Monte Carlo simulation mini-app coupling a traditional power iteration module with a time-dependent transport module, in a multi-group framework. Coyote supports slab geometries, with an arbitrary number of slabs associated to possibly dierent materials. Each material is dened by its energy-dependent macroscopic cross-sections and kinetic parameters. The boundary conditions can be periodic, reection, or vacuum. Coyote comes with a set of variance-reduction methods that can be freely and independently activated : Russian roulette and splitting, several versions of branchless collisions, forced ssion, implicit capture, forced decay of precursors, combined precursor, and importance ratios. Population control can be enforced by weight combing and sampling without replacement (in the power iteration module). The tallies are numerous and range from usual ux-based quantities to correlation functions, pair distance functions, Feynman moments of the ssion sites distribution, etc., as detailed in the following. Ensemble or ergodic averages can be computed, which is particularly useful for the analysis of correlations.

Première partie

Correlations in time-dependent diffusion problems solved by analog

Monte Carlo methods

-Space-time correlations in simplied kinetic simulations with neutrons and precursors

Kinetic Monte Carlo simulation lends itself quite naturally to the investigation of the statistical behaviour of stochastic neutron and precursor populations around their average. However, in order to establish a clear framework for our investigations, it is convenient to start our analysis from a model that is simple enough to obtain exact results, and yet suciently non-trivial to retain all the basic ingredients of realistic multiplying systems. This allows extracting the main trends, the scaling laws and the key guidelines that will be used in the following chapter for the interpretation of more complex time-dependent Monte Carlo simulations, including some elementary population-control mechanisms.

The aim of this chapter is thus to characterize the behaviour of space-time correlations and uctuations in a simplied model of nuclear reactor. For this purpose, instead of considering the (generally non-analog) Monte Carlo solution of the Boltzmann equation, we will introduce a diusion-based set of stochastic rules mimicking the neutron and precursors dynamics that can be sampled by analog Monte Carlo methods. The well known Galton-Watson birth-and-death stochastic process will be modied to account for precursors, which yields a two-species model with dierent transition rates for each species [START_REF] Harris | The Theory of Branching Processes[END_REF]. The modied Galton-Watson stochastic process will be coupled with a model of random spatial displacements within the diusion approximation, which under suitable assumptions can be derived from the exact neutron transport theory, as mentioned in Sec. 2.5.2. The combination of these two stochastic processes results in a Branching Brownian Motion (BBM), whose statistical moments will be shown to be attainable by analytical means. This is to be contrasted with the stochastic process underlying the Boltzmann equation, which is seldom amenable to closed-form solutions for its statistical moments. In the diusion framework, and relying mainly on the Pál-Bell backward formalism, we will be able to derive exact expressions for the characteristic quantities describing the behaviour of space-time correlations of neutrons and precursors in a nuclear reactor, and we will compare them to analog Monte Carlo simulation results obtained by directly sampling the BBM model, which will be used as a reference.

While our focus here is on the investigation of neutron uctuations in kinetic simulations, the use of analog Monte Carlo methods (where the sampling laws are directly inspired by the underlying physical process occurring in multiplying systems) allows immediately transposing the results obtained in this

Chapter to the analysis of neutron uctuations occurring in the real systems, up to a scaling factor due to the fact that the number of sampled particles in Monte Carlo simulations is usually much lower than the number of particles that are present in a nuclear reactor 1 .

Most of the material presented in this chapter is taken from our published work in Ref. 77.

. A simple stochastic model for nuclear reactor physics

In the simplest incarnation of our nuclear reactor model [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF], we represent the random movement of neutrons as a Brownian motion with a diusion coecient D, and we use a Galton-Watson birth-death process to describe ssion and capture. Each neutron undergoes sterile capture at a rate γ = 1/vΣ c and ssion events at a rate β = 1/vΣ f . Captured neutrons are simply killed. When ssion occurs, a random number of new neutrons are emitted at the position of the incident neutron, which is killed [START_REF] Williams | Random processes in nuclear reactors[END_REF]. The probability to emit instantaneously k prompt neutrons is denoted as p k . Delayed neutrons may further appear at the spatial site of a ssion event after a random, exponentially distributed time with rate λ, corresponding to the decay of the delayed neutron precursors [START_REF] Williams | Random processes in nuclear reactors[END_REF]. We denote by q k the probability that k delayed neutron precursors are created at a ssion event. For illustration, some representative neutron and precursor histories are shown in Fig. 3.1. In nuclear reactors, the average precursor decay time 1/λ is typically much larger than the average neutron lifetime 1/(γ + β). Furthermore, denoting respectively by ν p,1 = k kp k and ν d,1 = k kq k the average number of prompt and delayed neutrons resulting from ssion, we typically have ν p,1 ≫ ν d,1 [START_REF] Williams | Random processes in nuclear reactors[END_REF]. For reasons that will become clear in the following, we are using the notation ν p,l and ν d,l to indicate the l-th falling factorial moments of the number of neutrons and precursors produced from ssion, respectively, with

ν p,l = ∞ k=0 k(k -1) . . . (k -l + 1) p k (3.1)
and

ν d,l = ∞ k=0 k(k -1) . . . (k -l + 1) q k . (3.2)
For the sake of simplicity, our model neglects the energy dependence of the neutron-matter interaction rates. Furthermore, the spatial structure of the reactor core will be simplied by assuming that all the material properties (diusion coecient, ssion and capture rates) are homogeneous. All the physical parameters are taken to be uniform in space and constant in time. Contrary to Ref. 48, which considered unbounded systems, here the reactor will be modeled as a nite-size box with reection (Neumann) boundary conditions, mimicking the fact that in nuclear reactors neutron leakage from the core is deliberately kept small. To simplify matters even further, we will focus on one-dimensional systems, which are more easily amenable to analytical solutions : in what follows, we will thus consider a one-dimensional reactor of half-size L, i.e. the [-L, L] segment. This model captures the key physical mechanisms that are responsible for the uctuations of the ssion chains.

Throughout this chapter, we will use the parameter values given in Tab. 3.1, whose units are arbitrary.

. The critical regime

Nuclear reactors are operated at the critical regime, where the equilibrium between births by ssion and deaths by capture allows the neutron population and hence the heat production to be stationary on average [START_REF] Williams | Random processes in nuclear reactors[END_REF]. In our model, criticality is imposed by requiring that

β(ν p,1 + ν d,1 -1) = γ, (3.3)
which is equivalent to equating the production and disappearance rates. The criticality condition does not depend on the number of neutrons present in the system : indeed, nuclear reactors can be operated at

θ = 1 θ = 10 -1 θ = 10 -3 N 10 2 10 2 10 2 M 10 2 10 3 10 5 L 1 1 1 D 10 -2 10 -2 10 -2 β 0.2 0.2 0.2 γ 0.3 0.3 0.3 λ 10 -1 10 -2 10 -4 ν p,1 2 2 2 ν d,1 0.5 0.5 0.5 ν p,2 2 2 2 ν d,2 0 0 0 Table 3.1 : Different sets of parameters used in this work.
Here N is the initial (average) number of neutrons, M is the initial (average) number of precursors, L is the box half-size, D is the diffusion coefficient, β is the fission rate, γ is the capture rate, λ is the decay rate of precursors, ν p,l and ν d,l are the l-th falling factorial moments of the number of neutrons and precursors produced from fission, respectively, and θ = λ/(βν d,1 ). Units are arbitrary.

steady-state at virtually any power level, the power being proportional to the average number of neutrons in the core. Due to its inherently stochastic nature, the neutron population density at a spatial site will display uctuations (often dubbed `neutron noise' [START_REF] Williams | Random processes in nuclear reactors[END_REF]) because of random displacements in and out the spatial cell and because of random changes in the number of particles following ssion and capture events within the cell.

. Observables of interest

Fluctuations are characterized by the statistical moments of the neutron and precursor population as a function of position and time. Let V 1 and V 2 be two non-overlapping `detectors'2 located within the reactor. We are interested in computing the average number of neutrons E[n i ](t i ) and precursors E[m i ](t i ) present at time t i in detector i ∈ {1, 2}, and the two-point correlations between particles detected in V 1 at time t 1 and particles detected in V 2 at time t 2 , which stem from the cross-moments

E[n 1 n 2 ](t 1 , t 2 ), E[n 1 m 2 ](t 1 , t 2 ), and E[m 1 m 2 ](t 1 , t 2 ).
Often, it is preferable to work with continuous observables : we therefore dene the neutron and precursor densities by taking the limits

n(x i , t i ) = lim V i →0 E[n i ](t i ) V i (3.4) m(x i , t i ) = lim V i →0 E[m i ](t i ) V i , (3.5)
where x i is the center of the detector location, i ∈ {1, 2}. The average population sizes are then obtained by integrating the particle densities over the entire system :

n(t) = n(x, t) dx (3.6) m(t) = m(x, t) dx. (3.7)
Similarly, we dene the pair correlation functions by taking the limits

u(x 1 , t 1 , x 2 , t 2 ) = lim V 1 ,V 2 →0 E[n 1 n 2 ](t 1 , t 2 ) V 1 V 2 (3.8a) v(x 1 , t 1 , x 2 , t 2 ) = lim V 1 ,V 2 →0 E[n 1 m 2 ](t 1 , t 2 ) V 1 V 2 (3.8b) w(x 1 , t 1 , x 2 , t 2 ) = lim V 1 ,V 2 →0 E[m 1 m 2 ](t 1 , t 2 ) V 1 V 2 .
(3.8c)

The function u(x 1 , t 1 , x 2 , t 2 ) fully characterizes the behavior of uctuations and correlations of the neutron population, and in this respect is key to the investigation of neutron noise. Similarly as for the case of the average densities, integral quantities can be obtained by integrating the previous equations over the whole spatial domain : for the two-point neutron correlation function, e.g., we would have

u(t 1 , t 2 ) = u(x, t 1 , y, t 2 ) dx dy.
(3.9)

The information content of the two-point correlation function can be conveniently condensed in the mean-squared neutron pair distance ⟨r 2 ⟩(t), which is dened as

⟨r 2 ⟩(t) = (x 1 -x 2 ) 2 u(x 1 , t, x 2 , t) dx 1 dx 2 u(t, t) , (3.10)
which is also helpful in assessing the behaviour of neutron noise [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF].

. A theoretical framework for correlations analysis in the anarchic model

We begin our analysis by assuming that the neutron and precursor populations are free to evolve according to the stochastic rules described in Sec. 3.1, without any constraint. This system, which we will call the anarchic model (for illustration, see Fig. 3.1), is representative of an analog kinetic Monte Carlo simulation where no population control is imposed on the particle histories. The equations for the particle density and for the two-point correlation functions can be conveniently established using the backward master equation approach proposed by Pál and Bell (which is a special case of the Feynman-Kac backward formalism) [START_REF] Williams | Random processes in nuclear reactors[END_REF][START_REF] Pazsit | Neutron Fluctuations: A Treatise on the Physics of Branching Processes[END_REF]. The idea is to rst derive the equations for the moments n ′ i (x 0 , t 0 ) = n(x i , t i |x 0 , t 0 ), with i ∈ {1, 2}, and u ′ 1,2 (x 0 , t 0 ) = u(x 1 , t 1 , x 2 , t 2 |x 0 , t 0 ), conditioned to having a single initial neutron starting from position x 0 at time t 0 , treating x 1 , x 2 , t 1 and t 2 as parameters, and x 0 and t 0 as variables.

While the method is general, the derivation proposed below is original.

. Derivation of the (backward) moment equations for a single initial particle

Let P(n 1 , t 1 , n 2 , t 2 |x 0 , t 0 ) be the probability to nd n 1 particles at time t 1 in detector V 1 and n 2 particles at time t 2 in detector V 2 , for a single particle starting in x 0 at time t 0 , with t 2 > t 1 ; the results for t 2 < t 1 are recovered by arguments of symmetry. Pál and Bell have independently derived a backward formalism yielding the (adjoint) master equation for P. The manipulation of the resulting equation is rather cumbersome, and in most cases neutron noise can be characterized using the lowestorder moments of n 1 and n 2 . We will be in particular interested in the average number of neutrons in detector V 1 at time t 1 , namely,

E[n 1 ](t 1 |x 0 , t 0 ) = n 1 ,n 2 n 1 P(n 1 , t 1 , n 2 , t 2 |x 0 , t 0 ) (3.11)
and in the two-point correlation function between particles detected in V 1 at time t 1 and particles detected in V 2 at time t 2 , namely,

E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 ) = n 1 ,n 2 n 1 n 2 P(n 1 , t 1 , n 2 , t 2 |x 0 , t 0 ). (3.12)
By virtue of these considerations, it is therefore more convenient to introduce the probability-generating function associated to P, namely,

W (z 1 , z 2 , x 0 , t 0 ) = n 1 n 2 z n 1 1 z n 2 2 P (n 1 , n 2 , t 1 , t 2 |x 0 , t 0 ), (3.13)
where z i is the conjugate variable of n i , i ∈ {1, 2}, and hence derive the equations for the (falling factorial) moments, which follow from

E[n 1 (n 1 -1) . . . (n 1 -k+1)n 2 (n 2 -1) . . . (n 2 -l+1)](t 1 , t 2 |x 0 , t 0 ) = ∂ k+l ∂z k 1 ∂z l 2 W (z 1 , z 2 , x 0 , t 0 )| z 1 =1,z 2 =1 , (3.14)
with k, l ≥ 0 and the normalization W (1, 1, x 0 , t 0 ) = 1. It can be shown that W (z 1 , z 2 , x 0 , t 0 ) satises the non-linear backward evolution equation

- ∂ ∂t 0 W = D∇ 2 x 0 W -(γ + β)W + βG p (W )G d (W d ) + γ, (3.15)
where G p (z) = k z k p k is the probability-generating function associated to the prompt ssion probability distribution p k and G d (z) = k z k q k is the probability-generating function for the precursor probability distribution q k [START_REF] Pazsit | Neutron Fluctuations: A Treatise on the Physics of Branching Processes[END_REF]. The quantity W d in Eq. (3.15) is the probability-generating function associated to the creation of a precursor at (x 0 , t 0 ), namely,

W d (z 1 , z 2 |x 0 , t 0 ) = λ ∞ t 0 e -λ(t ′ -t 0 ) W (z 1 , z 2 |x 0 , t ′ )dt ′ . (3.16)
Taking the derivative of Eq. (3.15) once with respect to z 1 and using the denition in Eq. (3.14), we obtain the evolution equation for the average neutron number in V 1 , for a single starting particle in (x 0 , t 0 ) :

L † E[n 1 ](t 1 |x 0 , t 0 ) = 0, (3.17)
where L † is a backward linear operator dened by

L † f (x 0 , t 0 ) = ∂f ∂t 0 + D∇ 2 x 0 f + α p f + βν d,1 λ ∞ t 0 f (x 0 , t ′ ) e -λ(t ′ -t 0 ) dt ′ , (3.18)
with the shorthand

α p = β (ν p,1 -1) -γ. (3.19)
Equation (3.17) is to be solved with the nal condition

E[n 1 ](t 1 |x 0 , t 0 = t 1 ) = 1 if x 0 ∈ V 1 0 otherwise. ( 3

.20)

Letting G(x, t, x 0 , t 0 ) be the Green's function associated to the operator L † , the solution to Eq. (3.17

) reads E[n 1 ](t 1 |x 0 , t 0 ) = dx ′ χ V 1 (x ′ )G(x ′ , t 1 , x 0 , t 0 ), (3.21)
where χ V (x) is the characteristic function of a domain V . Taking the mixed derivative of Eq. (3.15) with respect to z 1 and z 2 , and using again the denition in Eq. (3.14), we obtain the evolution equation for the two-point correlation function, for a single particle starting in (x 0 , t 0 ) :

L † E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 ) = -f corr (t 1 , t 2 |x 0 , t 0 ), (3.22)
where we have dened

f corr (t 1 , t 2 |x 0 , t 0 ) = βν p,2 E[n 1 ]E[n 2 ] + βν p,1 ν d,1 E[n 1 ]λ t 2 t 0 e -λ(t ′ -t 0 ) E[n 2 ](t ′ )dt ′ + βν p,1 ν d,1 E[n 2 ]λ t 1 t 0 e -λ(t ′ -t 0 ) E[n 1 ](t ′ )dt ′ + βν d,2 λ 2 t 1 t 0 e -λ(t ′ -t 0 ) E[n 1 ](t ′ )dt ′ t 2 t 0 e -λ(t ′ -t 0 ) E[n 2 ](t ′ )dt ′ (3.23) with the shorthand notation E[n i ] = E[n i ](t 1 |x 0 , t 0 ), i ∈ {1, 2}. Equation (3.22) is to be solved with the nal condition E[n 1 n 2 ](t 0 = t 1 ) = E[n 2 ](t 2 |x 0 , t 0 = t 1 ) if x 0 ∈ V 1 0 otherwise.
(3.24)

Using the Green's function associated to the operator L † , the solution to Eq. (3.22) can be expressed as

E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 ) = dx ′ χ V 1 (x ′ )E[n 2 ](t 2 |x ′ , t 1 )G(x ′ , t 1 , x 0 , t 0 ) + t 1 t 0 dt ′ dx ′ f corr (t 1 , t 2 |x ′ , t ′ )G(x ′ , t ′ , x 0 , t 0 ). (3.25)
The continuous version of Eqs. (3.21) and (3.25) can be obtained by centering the detector regions V 1 and V 2 around x 1 and x 2 , respectively, dening

n(x 1 , t 1 |x 0 , t 0 ) = lim V 1 →0 E[n 1 ](t 1 |x 0 , t 0 ) V 1 u(x 1 , t 1 , x 2 , t 2 |x 0 , t 0 ) = lim V 1 ,V 2 →0 E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 ) V 1 V 2
and letting the detector sizes shrink to zero.

. Analysis of the Green's function of L †

By separation of variables, the Green's function G(x, t|x 0 , t 0 ) associated to L † can be expressed as

G(x, t, x 0 , t 0 ) = +∞ k=0 φ k (x)φ † k (x 0 )T k (t|t 0 ), (3.26)
where the time-eigenfunctions T k (t|t 0 ) read

T k (t|t 0 ) = ω + k -ω - k -1 × e ω + k (t-t 0 ) (ω + k + λ) -e ω - k (t-t 0 ) (ω - k + λ) , (3.27)
with two associated families of eigenvalues

ω ± k = α k + α p -λ ± (α k + α p + λ) 2 + 4λβν d,1 2 .
(3.28)

The quantities α k are the eigenvalues associated to operator D∇ 2

x , with the corresponding eigenfunctions φ k (x). For reection boundary conditions, we have α k = -D(kπ/2L) 2 and

φ k (x) = cos kπx 2L (3.29)
for k ≥ 0. The functions φ † k (x 0 ) are obtained from the orthonormality condition of the eigenfunctions, and for reection boundary conditions read

φ † k (x 0 ) =      1 2L if k = 0 1 L cos kπx 0 2L if k ≥ 1.
(3.30)

In nuclear reactors, the two families ω ± k are widely separated because of the typical values of λ, α p and βν d,1 . Loosely speaking, the ω + k family is associated to the dynamics of precursors, while ω - k describes prompt neutron dynamics.

. Generalization to a collection of particles

Once the single-particle moments are known, it is then straightforward to generalize the framework to describe a collection of N independent and identically distributed neutrons whose initial coordinates are distributed according to a given source Q(x 0 , t 0 ). Observe that the resulting probability-generating function W Q (z 1 , z 2 ) is related to the single-particle probability-generating function W (z 1 , z 2 , , x 0 , t 0 ) by

W Q N (z 1 , z 2 ) = t 1 -∞ dt 0 dx 0 W (z 1 , z 2 , x 0 , t 0 )Q(x 0 , t 0 ) N . (3.31)
Then, by taking the derivative of Eq. (3.31) once with respect to z 1 we obtain the average number of particles in V 1 for a collection of N source particles :

E[n 1 ](t 1 |Q N ) = N t 1 -∞ dt 0 dx 0 E[n 1 ](t 1 |x 0 , t 0 )Q(x 0 , t 0 ). (3.32)
Similarly, by taking the mixed derivative of Eq. (3.31) with respect to z 1 and z 2 we obtain the pair correlation between detector V 1 and V 2 for a collection of N source particles :

E[n 1 n 2 ](t 1 , t 2 |Q N ) = (N -1) N E[n 1 ](t 1 |Q N )E[n 2 ](t 2 |Q N ) + N t 1 -∞ dt 0 dx 0 E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 )Q(x 0 , t 0 ). (3.33)
In particular, if we denote the single-particle source distribution as Q(x 0 , t 0 ), the density n(x, t)

reads n(x, t) = N t -∞ Q(x 0 , t 0 )G(x, t|x 0 , t 0 ) dx 0 dt 0 , (3.34)
which expresses a linear superposition principle. The pair correlation function can be also explicitly computed based on the Green's function, and reads

u(x 1 , t 1 , x 2 , t 2 ) = N -1 N n(x a , t a ) n(x b , t b ) + G(x b , t b |x a , t a ) n(x a , t a ) + βν p,2 ta 0 G(x a , t a |x, t) G(x b , t b |x, t) n(x, t)dx dt + βν p,1 ν d,1 ta 0 G(x a , t a |x, t) c(x b , t b |x, t) n(x, t)dx dt + βν p,1 ν d,1 ta 0 G(x b , t b |x, t) c(x a , t a |x, t) n(x, t)dx dt + βν d,2 ta 0 c(x a , t a |x, t) c(x b , t b |x, t)n(x, t)dx dt, (3.35)
where

t a = min{t 1 , t 2 }, t b = max{t 1 , t 2 }, x a = x 1 if t a = t 1 x 2 if t a = t 2 , ( 3.36)
and

x b = x 1 if t b = t 1 x 2 if t b = t 2 .
(3.37)

The quantity

c(x a , t a |x, t) = λ ta t G(x a , t a |x, t ′ )e -λ(t ′ -ta) dt ′ (3.38)
represents the expected number of neutrons detected at (x a , t a ) produced by a precursor at (x, t), with t a > t.

The terms appearing in the correlation function in Eq. (3.35) can be given a physical interpretation.

The rst term describes the contribution of the detection of uncorrelated particles, i.e. particles that do not share any common ancestor. The second term describes self-correlations, i.e. the detection of the same particle history at (x a , t a ) and later at (x b , t b ). The four remaining terms represent the possible contributions due to ssion events : the rst one corresponds to the correlations induced by detection at (x a , t a ) and (x b , t b ) of neutrons stemming from dierent prompt neutrons produced by a common ssion event at (x, t) ; the following two terms are symmetrical and correspond to correlations induced by detection at (x a , t a ) and (x b , t b ) of neutrons stemming from a prompt neutron and a precursor produced by a common ssion event at (x, t) ; nally, the remaining term corresponds to correlations induced by detection at (x a , t a ) and (x b , t b ) of neutrons stemming from dierent precursors produced by a common ssion event at (x, t).

. The source corresponding to the critical regime

If the reactor is assumed to be in the critical regime, it is then possible to choose the single-particle source Q(x 0 , t 0 ) so that the average neutron density n(x, t) is stationary at any time t > 0 : this is for instance achieved by taking

Q(x 0 , t 0 ) = Q c (x 0 , t 0 ), with Q c (x 0 , t 0 ) = 1 2L θ 1 + θ δ(t 0 ) + 1 1 + θ λ exp(-λt 0 ) , (3.39)
where t 0 > 0 and θ = λ/(βν d,1 ). This particular choice of Q(x 0 , t 0 ) will be called the critical source.

The rst term in Eq. (3.39) corresponds to neutrons appearing at t 0 = 0 with probability θ/(1 + θ), while the second term corresponds to delayed neutrons appearing at exponentially distributed times from the decay of the precursors initially present at t 0 = 0, with complementary probability 1/(1 + θ). For our simple homogeneous one-dimensional reactor with reection boundary conditions, the spatial distribution 1/(2L) is at over the domain (Eq. (3.29)).

. Further generalizations of the Pál and Bell backward formalism

The Pál and Bell backward formalism lends itself to many possible generalizations. For instance, if we were to start our system with a collection of (independently distributed) N prompt neutrons, each having a source distribution Q n (x 0 , t 0 ), and M delayed neutrons, each having a source distribution Q m (x 0 , t 0 ), then Eq. (3.31) would be replaced by

W Q n+m (z 1 , z 2 ) = t 1 -∞ dt 0 dx 0 W (z 1 , z 2 , x 0 , t 0 )Q n (x 0 , t 0 ) N × t 1 -∞ dt 0 dx 0 W (z 1 , z 2 , x 0 , t 0 )Q m (x 0 , t 0 ) M . (3.40)
Correspondingly, setting N = N + M and denoting

Q n+m (x 0 , t 0 ) = N N + M Q n (x 0 , t 0 ) + M N + M Q m (x 0 , t 0 ), (3.41)
for the average number of particles we would have

E[n 1 ](t 1 |Q n+m ) = N t 1 -∞ dt 0 dx 0 E[n 1 ](t 1 |x 0 , t 0 )Q n+m (x 0 , t 0 ), (3.42)
and for the pair correlation we would have

E[n 1 n 2 ](t 1 , t 2 |Q n+m ) = N 2 E[n 1 ](t 1 |Q n+m )E[n 2 ](t 2 |Q n+m ) -N E[n 1 ](t 1 |Q n )E[n 2 ](t 2 |Q n ) -M E[n 1 ](t 1 |Q m )E[n 2 ](t 2 |Q m ) + N t 1 -∞ dt 0 dx 0 E[n 1 n 2 ](t 1 , t 2 |x 0 , t 0 )Q n+m (x 0 , t 0 ). (3.43)
Furthermore, the Pál and Bell backward formalism can be easily extended to include more general detectors (for instance, an array V = (V 1 , . . . , V K ) of K non-overlapping regions), and to more general sources (for instance, N and M can be themselves random variables with Poisson distributions).

u(x 1 , t 1 , x 2 , t 2 ) = N (N -1)θ 2 4L 2 (1 + θ) 2 + N θ 2L(1 + θ) G(x 2 , t 2 , x 1 , t 1 ) + N θ 2L(1 + θ) [u pp (x 1 , t 1 , x 2 , t 2 ) + u pd (x 1 , t 1 , x 2 , t 2 ) + u dp (x 1 , t 1 , x 2 , t 2 ) + u dd (x 1 , t 1 , x 2 , t 2 )] . (3. 44
)
Each component is dened in terms of the respective eigenmode expansion, namely,

u pp = k φ k (x 1 )φ † k (x 2 )u k pp (3.45) u pd = k φ k (x 1 )φ † k (x 2 )u k pd (3.46) u dp = k φ k (x 1 )φ † k (x 2 )u k dp (3.47) u dd = k φ k (x 1 )φ † k (x 2 )u k dd .
(3.48)

For the prompt-prompt component u k pp we have :

u 0 pp (t 1 , t 2 ) = βν p,2 (1 + θ) 2 θ 2 t 1 + θ e ω d t 1 -1 ω d + e ω d t 2 -e ω d (t 2 -t 1 ) ω d + e ω d (t 1 +t 2 ) -e ω d (t 2 -t 1 ) 2ω d , (3.49)
where we have introduced ω d = -τ -1 2 = -βν d,1 -λ, and

u k pp (t 1 , t 2 ) = βν p,2 (ω + k -ω - k ) 2 (ω + k + λ) 2 2ω + k e ω + k (t 1 +t 2 ) -e ω + k (t 2 -t 1 ) + (ω - k + λ) 2 2ω - k e ω - k (t 1 +t 2 ) -e ω - k (t 2 -t 1 ) + (ω + k + λ)(ω - k + λ) ω + k + ω - k e ω + k (t 2 -t 1 ) + e ω - k (t 2 -t 1 ) -(e ω + k t 1 +ω - k t 2 + e ω + k t 2 +ω - k t 1 ) , for k ≥ 1. For the prompt-delayed component u k pd we have u 0 pd (t 1 , t 2 ) = βν d,1 ν p,1 θ (1 + θ) 2 θt 1 + e ω d t 1 -1 ω d + θ(e ω d (t 2 -t 1 ) -e ω d t 2 ) ω d + e ω d (t 2 -t 1 ) -e ω d (t 1 +t 2 ) 2ω d (3.50) and u k pd (t 1 , t 2 ) = βλν d,1 ν p,1 (ω + k -ω - k ) 2 (ω + k + λ) e ω + k (t 1 +t 2 ) -e ω + k (t 2 -t 1 ) 2ω + k + e ω - k (t 2 -t 1 ) -e ω + k t 1 +ω - k t 2 ω + k + ω - k + (ω - k + λ) e ω + k (t 2 -t 1 ) -e ω + k t 2 +ω - k t 1 ω + k + ω - k + e ω - k (t 1 +t 2 ) -e ω - k (t 2 -t 1 ) 2ω - k (3.51) for k ≥ 1. For the delayed-prompt component u k dp we have u 0 dp (t 1 , t 2 ) = βν d,1 ν p,1 θ (1 + θ) 2 θt 1 + θ 1 -e ω d t 1 ω d + e ω d t 2 -e ω d (t 2 -t 1 ) ω d + e ω d (t 1 -t 2 ) -e ω d (t 2 +t 1 ) 2ω d (3.52) and u k dp (t 1 , t 2 ) = βλν d,1 ν p,1 (ω + k -ω - k ) 2 (ω + k + λ) e ω + k (t 1 +t 2 ) -e ω + k (t 2 -t 1 ) 2ω + k + e ω + k (t 2 -t 1 ) -e ω + k t 2 +ω - k t 1 ω + k + ω - k + (ω - k + λ) e ω - k (t 2 -t 1 ) -e ω + k t 1 +ω - k t 2 ω + k + ω - k + e ω - k (t 1 +t 2 ) -e ω - k (t 2 -t 1 ) 2ω - k (3.53)
for k ≥ 1. Finally, for the delayed-delayed component u k dd we have (3.44).

u 0 dd (t 1 , t 2 ) = βν d,2 θ 2 (1 + θ) 2 t 1 + 1 + e ω d (t 1 -t 2 ) -e ω d t 1 -e ω d t 2 ω d + e ω d (t 1 +t 2 ) -e ω d (t 2 -t 1 ) 2ω d (3.54) and u k dd (t 1 , t 2 ) = βλ 2 ν d,2 (ω + k -ω - k ) 2 e ω + k (t 1 +t 2 ) -e ω + k (t 2 -t 1 ) 2ω + k + e ω - k (t 1 +t 2 ) -e ω - k (t 2 -t 1 ) 2ω - k + e ω + k (t 2 -t 1 ) + e ω - k (t 2 -t 1 ) -e ω + k t 1 +ω - k t 2 -e ω - k t 1 +ω + k t 2 ω + k + ω - k (3.
Taking t 1 = t 2 = t, we can obtain an expression for the mean-squared pair distance ⟨r 2 ⟩(t). For our case (reection boundary conditions, ν d,2 = 0), the expression reads : 4 (3.56)

⟨r 2 ⟩(t) = (x -y) 2 u(x, y, t) dx dy u(t, t) = 2L 2 3 1 - 1 N + u pp 0 (t) + u dp 0 (t) + u 0 pd (t) - 64L 2 N + u 0 pp (t) + u 0 dp (t) + u 0 pd (t) +∞ k=1 k odd u k pp (t) + u k dp (t) + u k pd (t) (kπ)

. Analysis of the pair correlation function

Inspection of Eq. (3.34) shows that the average neutron density in a critical reactor starting form the critical source in Eq. (3.39) is trivially constant, as expected, and reads n(x, t) = N/(2L). This value depends on the number N of neutrons emitted from the source and on the reactor size 2L. Correspondingly, the average total number of neutrons n(t) will be constant and equal to N , and it can be shown that the average total number of precursors m(t) will be also constant and equal to M = N/θ.

The correlation function in Eq. (3.35), on the contrary, is not stationary for a critical system. While Eq. (3.35) is exact, the evaluation of G requires the numerical computation of an innite series. In the following, the semi-analytical solution is henceforth obtained using Eqs.

ũ(x 1 , t 1 , x 2 , t 2 ) = u(x 1 , t 1 , x 2 , t 2 ) -δ(x 1 -x 2 ) n(x 1 , t 1 )
, which is equivalent to ignoring the singular term δ(x 1 -x 2 ) due to the self-correlations occurring in the system at t 1 = t 2 .

Note that this does not remove all of the self-correlation contributions, as self-correlations also include contributions from points x 1 ̸ = x 2 . By abuse of language, and when it does not impede understanding, in the following we shall call ũ the `pair correlation function' nonetheless. Since the function has four (a) On the graph we plot u(x 1 , t 1 , x 2 , t) for x 1 = x 2 = 0 and t 1 = T /2. The bottom graph shows u(x 1 , t 1 , x, t 2 ) the spatial shape when t 1 = t 2 = T /2 and x 1 = 0. top panel) and 3.2b (top panel) show that, after an initial transient, t ≲ 50, the time prole of the pair correlation function exhibits a linear build-up. For t 1 > t 2 , correlations saturate. Indeed, the linear build-up describes ssions happening at time t 1 that may contribute to further ssions occurring at a xed t 2 , meaning that obviously ssions occurring at t 1 > t 2 cannot contribute.

The analysis of u(x 1 , t 1 , x 2 , t 2 ) shows that the correlation function displays several typical time scales. First, the spatial boundedness of the system introduces timescales that are related to diusion.

The prompt time scales accumulate around and are asymptotically dominated by the mixing time, which is related to the time a neutron needs in order to explore the whole conguration space [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF]. With the introduction of precursors, additional remarkable timescales appear. As shown in Eq.

time constants 2ω + k , 2ω - k , ω + k + ω - k and ω + k -ω - k (see Sec. 3.2.
2 for details). If we restrict our attention to the one-time, two-point correlation function, u(

x 1 , x 2 , t) = u(x 1 , t, x 2 , t), then the ω + k -ω - k modes
disappear. The rst few characteristic time constants for each family are represented in Fig. 3.3. We can identify four dierent interesting timescales :

• the separation of the rst two characteristic time constants of the ω + k + ω - k family is given by

(ω + 0 + ω - 0 ) -(ω + 1 + ω - 1 ) = -α 1 = 4L 2 /Dπ 2 = τ -1
D . This is exactly the inverse of the mixing time for a system without delayed neutrons, as studied in Ref. [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF]. For the parameter values of • on a timescale of the order of τ

2 = |ω + 0 + ω - 0 | -1 = (βν d,1 + λ) -1
, the correlation function reaches a quasi-stationary regime, after prompt dynamics has stabilized but before delayed dynamics comes into play. Note that the expression of τ 2 is independent of the spatial characteristics of the system, such as its spatial extent or the diusion coecient. Therefore, this time scale corresponds to a collective behaviour. For times of the order of a few τ 2 , the prompt modes contribute a time-independent term to the correlation function. The delayed modes, on the other hand, have timescales that are much longer than τ 2 ; therefore, the remaining exponentials terms exp(2ω

+ k t)
can be replaced with their rst-order Maclaurin development, yielding a linear dependency on time.

This explains the presence of linear build-up regimes in Figs. [START_REF]Neutronics and thermal-hydraulic coupling in LWR technology[END_REF].2a and 3.2b. For the parameters of these gures, we have τ 2 ≃ 9.

• on a timescale of the order of τ 1 = |2ω + 1 | -1 , the correlation function acquires its asymptotic spatial shape. In this sense, this timescale is analogous to the mixing time of prompt systems ; however, given the expression of Eq. (3.28), it involves the precursor decay constant λ and is therefore generally much longer than the prompt mixing time, which is given by τ D . For the parameter values of Figs. 3.2a and 3.2b, we have τ 1 ≃ 269.

• for times much longer than τ 1 , all the exponential terms in the correlation function have died out.

The spatial shape of the correlation function is frozen, but the overall scale factor increases linearly with t. It has been previously shown by Houchmandzadeh et al. [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF] (and our calculations conrm this) that the slope of this eventual linear regime is τ -1 E , where

τ E = N (1 + θ) 2 βν 2 θ 2 , (3.57)
where

ν 2 = ν p,2 + 2ν d,1 ν p,1 + ν d,2 ,
and N = N θ/(1 + θ) is the average number of neutrons in the system. After a time of the order of τ E , the standard deviation of the total population size becomes equal to the mean population size. This is the regime of the critical catastrophe : because of the unbounded uctuations, the population will almost surely go eventually extinct, despite the fact that the average population size is constant [START_REF] Williams | Random processes in nuclear reactors[END_REF][START_REF] Pazsit | Neutron Fluctuations: A Treatise on the Physics of Branching Processes[END_REF]. This apparently paradoxical behaviour is a well-known characteristic of critical birth-and-death processes. The quantity τ E thus physically represents the typical extinction time of the system. For the parameter values of Figs. 3.2a and 3.2b, the extinction time is τ E ≃ 1.5 × 10 4 ; therefore, this regime is not observable in the gures above.

The space prole of the correlation function, shown in Fig. 3.2a (bottom panel), has the tent-like shape that is typical of clustering behaviour. Despite the fact that the mean population density is uniform and constant, specic histories can exhibit strong patchiness (clusters) where pairs of particles will tend to be located at short distances from each other. This is coherent with previous ndings for a reactor model neglecting delayed neutrons [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF][START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF]. Figure 3.2b (bottom) suggests a relaxation toward a at distribution for t 1 and t 2 suciently far from each another. Further analyses (not shown here for conciseness) conrm this conjecture.

It is tempting to characterize the importance of neutron clustering by dening a dimensionless parameter η = τ 1 /τ E , i.e. the ratio of the two longest timescales in the system. When η ≫ 1, clustering will relax slowly over a time of the order of the extinction time, and therefore it will be observable over the whole lifetime of the system. It is interesting to consider how the dimensionless parameter η scales with θ for small θ, which is the regime of interest for nuclear reactors. By replacing λ = θβν d,1 in Eq. (3.28) and developing in Maclaurin series in θ, we obtain

ω + 1 ≃ θ α -1 1 + α -1 p . (3.58)
Hence,

τ 1 ≃ α -1 1 + α -1 p 2θ . (3.59)
This should be contrasted with the mixing time for a system without delayed neutrons, which is τ D = 1/α 1 . Even in the regime where diusion dominates (α 1 ≪ α p ), we can see that the presence of delayed neutrons slows down mixing by a factor of 1/θ. Unsurprisingly, this is the factor that is introduced by delayed neutrons in the characteristic response times of the mean reactor populations around criticality [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF]. Thus, mixing is essentially a phenomenon that characterizes the behaviour of the average neutron and precursor densities.

When Eq. (3.59) is replaced in the denition of the clustering parameter η, we nd

η = τ 1 τ E ≃ α -1 1 + α -1 p 2N βν 2 θ. (3.60)
This goes to show that the presence of delayed neutrons is very eective in quenching clustering, because of the dierent scaling of τ 1 and τ E with θ. It is worth remarking in passing that the fact that the extinction time scales as θ -2 for small values of θ is not a coincidence, but it is due to the fact that τ E is a purely stochastic quantity, which can only be dened in terms of the second-order moments of the populations.

. Analysis of the mean-squared pair distance

The quantity ⟨r 2 ⟩(t) yields the average square distance between pairs of particles (observed at the same time), and in this respect provides information on their tendency toward clustering [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. At time t 0 = 0, we have

⟨r 2 ⟩(0) = ⟨r 2 ⟩ iid = 2L 2 3 , (3.61)
since the neutrons are independently and uniformly distributed within the reactor. At later times, neutrons exhibit correlations induced by ssion events and correspondingly ⟨r 2 ⟩(t) < ⟨r 2 ⟩ iid , which is a signature of the spatial clustering regime. For the anarchic model, we can obtain semi-analytical formulas for ⟨r 2 ⟩(t) in the form of a series, whose expression can be found in Eq. (3.56), generalizing previous ndings for the model without delayed neutrons [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. neutron pairs present in the system at a given time, normalized by the ensemble average of the square of the number of particles at a given time. For times of the order of the extinction time, the quantity ⟨r 2 ⟩(t) asymptotically reverts to the uncorrelated value ⟨r 2 ⟩ iid , with a convergence speed that depends on the model parameters. This is explained by the fact that for t ≫ τ E the population in a critical regime almost certainly dies out and correlations become spatially at over the entire reactor. This result is again coherent with previous ndings for the case where precursors were neglected [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

. Conclusions

We considered a simplied one-dimensional stochastic model of a nuclear reactor including neutrons and precursors. We derived the moment equations characterizing the time and space correlations of the particles present in the system.

Our main tool was the two-point, two-time neutron-neutron second-order moment (the pair correlation function), which characterizes the joint observation of neutrons at (x 1 , t 1 ) and at (x 2 , t 2 ). By using the Pál-Bell backward formalism, we derived the exact pair correlation function for the anarchic model, where all the populations are left free to evolve, starting from a distributed source composed of a collection of initial particles (neutrons, precursors or both).

We identied two typical timescales, namely the time τ 2 necessary for the prompt correlation dynamics to relax, and the time τ 1 necessary for correlations to achieve their asymptotic spatial shape. The latter generalizes the concept of mixing time (initially introduced for purely prompt stochastic models, without precursors) to a system with delayed neutrons. For typical values of the physical parameters, τ 1 is actually much longer than the mixing time of a critical system without delayed neutrons.

The system is further characterized by its extinction time τ E , which represents the typical time required to observe catastrophic uctuations of the total population size. Similarly to the mixing time, the extinction time for a system with delayed neutrons is much longer than the extinction time for a critical system without delayed neutrons, for typical values of the physical parameters.

-Space-Time Correlations in simplied kinetic simulations with idealized population control

So far, we have considered the case of kinetic simulations using freely evolving particle populations.

In practice, some forms of population control will be generally used, in order to prevent fatal numerical instabilities such as the `critical catastrophe' discussed in the previous chapter.

In the following, we will introduce and thoroughly examine a class of stochastic models coupling the branching Brownian model presented in Chapter 3 with an idealized mechanism of population control that mimics more realistic strategies, but is simple enough to allow for analytical solutions for the observables of interest.

For this purpose, we will suitably generalize an approach originally conceived for the case where precursors are neglected [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. For this model, a control mechanism has been introduced as follows :

whenever a neutron is created by ssion, another neutron is randomly chosen and eliminated [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF].

This sampling strategy was inspired by heuristic arguments proposed in the context of theoretical ecology [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF][START_REF] Zhang | Diusion reproduction processes[END_REF] : it preserves exactly the total number of individuals in the system and has a deep impact on the behaviour of the correlations. In particular, it has been shown to induce an upper bound on the amplitude of neutron noise and thus prevent the occurrence of the critical catastrophe [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF].

The enforcement of control mechanisms, which correlate birth and death events, breaks the independence of particle histories, so that the backward formalism can no longer be used [START_REF] Pál | Neutron uctuations in a multiplying medium randomly varying in time[END_REF]. We will resort to a forward master equation approach instead, which is slightly more involved. Thus, for the sake of simplicity we will focus exclusively on the case of binary ssion and on the production of at most one delayed neutron precursor, although our ndings can be straightforwardly extended to general neutron and precursor production distributions, at the expense of more cumbersome formulas. For the same reason, contrary to Chapter 3, we shall only consider spatial correlations with both particles observed at the same time.

The analysis of spatial moments will be complemented by the investigation of the statistical behaviour of neutron genealogical trees. The main idea behind this approach is that ssion progeny originating from the same ssion event are correlated and share a common ancestor. Neutrons sharing at least one common ancestor form a genealogical tree of correlated particles, which we call family by analogy with population dynamics. We will show that, in the course of the stochastic evolution of the population dynamics obeying the rules of the Galton-Watson birth-and-death process with population control, all the families but one will eventually become extinct [START_REF] Serva | On the genealogy of populations: trees, branches and ospring[END_REF]. The relationship between the evolution of the statistics of the neutron families and spatial correlations is encoded in the competition between birth, death, and diusion. In this chapter, we will provide a formal framework to derive analytically the moments of the distribution of the extinction time for all the families but one, for a collection of neutrons and precursors evolving in time under dierent population control mechanisms.

Population control has a clear physical counterpart in real systems. In nuclear reactors, various feedback mechanisms act against the excursions of the neutron population. An example of utmost importance is provided by the Doppler eect. The probability of neutron-matter interaction depends on the temperature of the medium ; in nominal conditions, the probability of sterile absorption increases with increasing temperature [2]. Since the temperature of the reactor is driven by the number of neutrons in the system, the overall result of the Doppler eect is that deviations from the mean in the number of neutrons in the reactor tend to be quenched. The typical timescale of the Doppler eect is shorter than the neutron lifetime, so that the feedback acts almost instantaneously on the behaviour of the ssion chains [2]. In this respect, the analysis of analog kinetic Monte Carlo simulations with population control can be transposed to the analysis of neutron uctuations occurring in the real systems when feedback mechanisms are taken into account.

Most of the work presented in this chapter has been previously published in Ref. [START_REF] Bonnet | Space and time correlations for diusion models with prompt and delayed birth-and-death events[END_REF] and Ref. [START_REF] Bonnet | The statistics of family histories in kinetic Monte-Carlo simulations[END_REF].

. Application of population control to a collection of neutrons alone

An extensive characterization of the correlations for a collection of diusing and multiplying particles where the number of individuals is kept exactly constant has been derived in the literature [START_REF] De Mulatier | The critical catastrophe revisited[END_REF], building upon previous results from theoretical ecology [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. In these works, precursors were neglected. Moreover, the approach presented in Ref. [START_REF] De Mulatier | The critical catastrophe revisited[END_REF] does not lend itself to a straightforward generalization to the broader case of a collection of neutrons and precursors.

To overcome this issue, we introduce here a novel strategy that allows deriving a rigorous analytical framework based on forward master equations. For pedagogical reasons, we begin our investigation by considering the simpler case of a collection of neutrons alone, where our approach recovers the results of Ref. [START_REF] De Mulatier | The critical catastrophe revisited[END_REF], albeit in a more general setting that can be then extended to take into account the presence of precursors. This new model will be discussed in Sec. 4.2 and following.

If precursors are neglected, a stochastic model with a population control constraint that preserves exactly the number of neutrons in the system can be obtained by setting the rules dened in Ref. [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

Each neutron induces ssion with rate β, producing exactly two prompt neutrons. Capture is conditioned to ssion : upon ssion, another neutron randomly chosen in the remaining population is eliminated.

Between ssion events, neutrons undergo diusion. This ensures that the number of neutrons N is constant, which corresponds to a critical reactor with strict population control.

. Derivation of the space-dependent master equation

We assume again that the viable space is the one-dimensional domain [-L, L]. Following the strategy introduced in Ref. [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF], we derive the forward space-dependent master equation by partitioning the domain into a set of K segments of equal length δx = 2L/K, which we denote as V k with k ∈ {1, . . . , K}.

Correspondingly, we write n = (n 1 , . . . , n K ). Let P(n, t) be the joint probability observing n k neutrons in each of the K detectors, with k ∈ {1, . . . , K}, given an initial condition at t 0 = 0 of N neutrons uniformly distributed in space. We have i n i = N . In addition, for conceptual ease, diusion will be rst modelled as a discrete random walk by dening the rate ξ at which a neutron jumps from V k to V k±1 . The evolution equations for P(n, t) can be written more concisely introducing the annihilation a k and creation a † k operators, whose action on a state vector v = (v 1 , . . . , v K ) is dened by

a k v = (. . . , v k-1 , v k -1, v k+1 , . . . ) a † k v = (. . . , v k-1 , v k + 1, v k+1 , . . . ).
The stochastic rules allow for the following transitions

• a transition n → a i a † i±1 n with rate per neutron ξ, corresponding to diusion from site i to neighbouring sites.

• a transition n → a † i a j n with rate per neutron β and probability

p = n i -1 N n j + 1 N -1 , (4.1)
corresponding to ssion in bin i and capture in bin j.

Based on these rules, the discrete master equation reads (4.

∂ ∂t P(n, t) = i   ξ(n i+1 + 1)P(a † i+1 a i n, t) + ξ(n i-1 + 1)P(a † i-1 a i n, t) -2ξn i P(n, t) + βN j̸ =i n i + 1 N n j -1 N -1 P(a † i a j n, t) - n i N n j N -1 P(n, t)   . ( 4
3)

The two-point correlation between neutrons detected in V i and neutrons detected in V j at time t is similarly derived from

E[n i n j ](t) = n n i n j P(n, t).
(4.4)

The equation for the spatial moment of order l can be obtained by multiplying Eq. (4.2) by n l k and summing over all possible states. For the rst moment, this procedure yields

∂ ∂t E[n k ](t) = γ∆ k E[n k ](t), (4.5)
where ∆ k denotes the discrete Laplace operator, such that

∆ k f k = f k+1 -2f k + f k-1 . Taking the limit V i → 0 leads to the one-dimensional diusion equation ∂ ∂t n(x, t) = D∇ 2 n(x, t), (4.6)
with V i centered around x. The diusion rate ξ is assumed to scale as 1/ √ δx, so that the diusion coecient is recovered as

lim δx→0 ξδx 2 = D. (4.7)
With reection boundary conditions, the solution of Eq. (4.6) is simply the spatially uniform density n(x, t) = N 2L .

(4.8)

Similarly, for the second moment we obtain

∂ ∂t E[n k n l ](t) = γ (∆ l E[n k n l ](t) + ∆ k E[n l n k ](t)) - 2β N 0 -1 E[n k n l ](t) + δ k,l 2N 0 β N 0 -1 E[n k ](t) + δ i,j ξ (E[n i+1 ](t) + E[n i-1 ](t) + 2E[n i ](t)) -δ i+1,j ξ (E[n i+1 ](t) + E[n i ](t)) -δ i-1,j ξ (E[n i ](t) + E[n i-1 ](t)) , (4.9)
where δ i,j denotes the Kronecker symbol. Taking again the limit V i → 0 leads to

∂ ∂t ũ(x, y, t) = D(∇ 2 x + ∇ 2 y )ũ(x, y, t) - 2β N -1 ũ(x, y, t) + 2βn(x, t)δ(x -y), (4.10)
where, for the sake of conciseness, we have introduced the modied pair correlation function ũ(x, y, t) = u(x, y, t) -δ(x -y)n(x, t). Here δ denotes the Dirac distribution. The detectors V i and V j are centered around x and y, respectively. Finally, we assume lim

V i ,V j →0 δ i,j = δ(x -y).
(4.11)

When taking the limit V i → 0, the last two lines of Eq. (4.9) lead to higher-order derivatives of the Dirac distribution, that disappear when using ũ instead of u.

Equation (4.10) is a linear equation that can be solved in terms of the Green's function

G(x, y, t|x ′ , y ′ , t ′ ) = G(x, t|x ′ , t ′ )G(y, t|y ′ , t ′ )e -2β N -1 (t-t ′ ) , (4.12)
where G is the Green's function associated to Eq. (4.6), i.e. the Gaussian propagator, which read

G(x, t|x ′ , t ′ ) = 1 4πD(t -t ′ ) e (x-x ′ ) 2 4D(t-t ′ ) .
(4.13)

The formal solution of Eq. (4.10) then reads

ũ(x, y, t) = 2β dx ′ dt ′ G(x, t|x ′ , t ′ )G(y, t|x ′ , t ′ )e -2β N -1 (t-t ′ ) n(x ′ , t ′ ) + N (N -1) N 2 n(x, t)n(y, t)e -2β N -1 t . ( 4

.14)

The exponential term in Eq. (4.14) indicates that correlations saturate exponentially fast, with a typical timescale τ M = (N -1)/2β, instead of diverging linearly as in the anarchic model, and the critical catastrophe is consequently averted. Spatial clustering is therefore mitigated because correlations now admit a maximum value. These results are consistent with the ndings previously derived using the heuristic arguments of Ref. 28 (see also Ref. [START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF]). In addition, contrary to the backward formalism which relies on the assumption of statistical independence between particles, the forward formalism developed here can take into account the particle correlations arising due to mass conservation.

. Analysis of the statistics of neutron families

The two-point function ũ(x, y, t) encodes the spatial correlations of a collection of neutrons, under the constraint that the total population N is kept constant. A complementary investigation of the statistical behaviour of the individuals in birth-and-death models can be carried out by determining the distribution of the number of neutron `families' as a function of time : this approach has been rst suggested in the context of theoretical biology [START_REF] Zhang | Diusion reproduction processes[END_REF]8587], then applied to reactor physics by de Mulatier et al. [START_REF] De Mulatier | The critical catastrophe revisited[END_REF],

and further investigated by Sutton and Mittal for the analysis of clustering phenomena in Monte Carlo simulations [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF].

Consider again the time-dependent Galton-Watson birth-and-death process in which neutrons reproduce upon ssion, and die upon capture, in the one-dimensional reected system introduced in the previous section.

Source neutrons are assigned an identier in {1, . . . , F }, which they transmit to their descendants.

Let n i be the number of neutrons with identier i. Neutrons sharing the same identier form a family. The simulation starts with F initial families. The size of a family increases by one when one of its members induces ssion ; it decreases by one when a neutron is the family is killed by capture. A family becomes extinct when the last of its members is captured. Some families will thus disappear over time, while others will proliferate, inducing a large number of descendants. If no additional families are introduced, which in reactor physics is equivalent to the absence of either an external source or spontaneous ssion, all the families but one will eventually become extinct, leading to the xation of the neutron population [START_REF] Serva | On the genealogy of populations: trees, branches and ospring[END_REF]. The time to xation is called xation time : we will show that this quantity may be related to the timescale driving the growth and saturation of correlations. The xation time in mass-preserving models somehow plays the role of the extinction time in the anarchic model, with a signicant dierence : the moments of the extinction time are not nite, whereas those of the xation time are nite, and will be investigated in the remainder of this section.

In order to shed light on the evolution of neutron families, we will derive the equations governing the rst few moments of the family statistics. Here, we present the derivation of the xation time for a population of prompt neutrons in a homogeneous medium, in a continuous-time context and neglecting precursors, and show that we can obtain closed-form expressions for these quantities.

f p (n)E [τ p |A] (n) = p + f p (n + 1)E [(τ p + δτ (n))|A] (n + 1) +p -f p (n -1)E [(τ p + δτ (n))|A] (n -1), (4.15)
where p ± are the probabilities that transition n → n ± 1 happens. In this context, we simply have p + = p -= 1/2, while δτ (n) = (N -1)/(βn(N -n)) is the inverse of the total rate of `useful' transitions, i.e., transitions that change the number of neutrons in A.

Equation (4.15) can be recast in a concise form by dening τp,

1 (n) = nE[τ p |A](n) if n > 0 and τp,1 (0) = 0, yielding τp,1 (n + 1) -2τ p,1 (n) + τp,1 (n -1) = - N -1 β(N -n) , (4.16)
which is to be solved with the boundary condition τp,1 (N ) = 0 (i.e., the population is already xed if n = N ), and τp,1 (0) = 0 by construction. We obtain the explicit solution The expression for E[τ p |A](n) yields the conditional average xation time for a family of initial size n. We dene then τ p (n 1 , . . . , n F ) as the average xation time for a population consisting of competing families with initial sizes n 1 , . . . , n F , which is given by the average of the family-wise conditional xation time :

E[τ p |A](n) = N -1 βn [1 -n + (N -n)(H N -1 -H N -n )] + (N -1) 2 βN , (4.17
τ p (n 1 , . . . , n F ) = i=1,...,F f (n i )E[τ p |A](n i ). (4.18)
In particular, when N = F and n i = 1 ∀i, then τ p (1, . . . , 1) = E[τ p |A](1) = (N -1) 2 /(βN ).

Let us now examine the variance of the xation time, using the law of total expectation on E[τ 2 p |A](n). After dening τp,2 (n) = nE[τ 2

p |A](n) for n > 0 and τp,2 (0) = 0, we nd that τp,2 (n) satises the equation

τp,2 (n + 1) -2τ p,2 (n) + τp,2 (n -1) = (N -1) β(N -n) 1 βN -2E[τ p |A](n) , (4.19)
E[τ 2 p |A](n) = τ S p,2 (n) -E[τ p |A](n) + C N , (4.20)
where τ S 2 (n) and C N are given by

τ S p,2 (n) = 2N N -1 βN 2   N n n-1 j=1 H N -j (n -j) j -[1 + N H N -1 ] (H n -1)   C N = N -1 βN 2    2N   (H N -1)(1 + N H N -1 ) - N -1 j=1 H N -j (N -j) j   -1    .
It is worth noting that the standard deviation σ[τ p ](n) = Var[τ p |A] of the xation time is of the same order of magnitude as the average xation time : in some cases, xation might happen relatively early in the simulation. In particular, the ratio of σ[τ p ](n) over E[τ p ](n) in Fig. 4.2 shows that the latter is of the same order of magnitude as the former, even for n ≪ N .

It is interesting to remark that, due to our choice of mass-preserving boundary conditions, the statistics of families for the bounded box reactor is insensitive to spatial eects : the moments of the xation time do not depend on the typical spatial scale of the problem at hand. This is in stark contrast with the two-point correlation function for the same system, and thus with the squared pair distance ⟨r 2 (t)⟩. For the case of particles evolving with population control in unbounded homogeneous media, Meyer et al.

showed that ⟨r 2 (t)⟩ reads

⟨r 2 (t)⟩ = 4Dτ R,p 1 -e -t τ R,p , (4.21)
indicating that the saturation occurs over a characteristic timescale τ R,p = (N -1)/(2β) [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. It follows that for N ≫ 1 we have τ R,p ∼ E[τ p |A](1)/2, i.e., the average xation time is equal to twice the renewal time, which suggests a close relation between the statistical behaviour of families and the one of spatial clustering [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. However, for particles evolving with population control in bounded homogeneous media having reective boundary conditions, the exponential relaxation of ⟨r 2 (t)⟩ towards an asymptotic value does depend on the spatial scale of the system (i.e. the mixing time) [START_REF] De Mulatier | The critical catastrophe revisited[END_REF], whereas the moments of the xation time do not, as shown here. The identication of xation and clustering timescales appears thus to be fully meaningful for innite and homogeneous media alone. Indeed, the emergence of clustering in conned geometries results from the competition between xation and spatial mixing due to diusion [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF] : the xation time cannot be identied with the clustering timescale anymore.

. Control of neutrons in a system of neutrons and precursors

We will now revisit the stochastic model introduced in Sec. 4.1 by introducing precursors : we will show that the eects of population control become more cumbersome to take into account ; nonetheless, we will be able to derive a class of models that allow for analytical or semi-analytical solutions that can be benchmarked against Monte Carlo simulations.

Let us consider a population of neutrons and precursors, evolving based on the stochastic rules described in Sec. 3.2. Since neutrons are responsible for the Monte Carlo tallies, it seems natural to enforce population control on neutrons alone, which can be achieved as follows.

Each neutron induces ssion with a rate of β ; the ssion event produces exactly two prompt neutrons (with probability p 2 = 1). Additionally, ssion events produce a precursor with probability q 1 , and no precursor with the complementary probability q 0 = 1 -q 1 . Upon decay, with rate λ, the precursor yields a delayed neutron. In this specic case, we have q 1 = ν d,1 , which is not true in general. In order to ensure a constant number N of neutrons in the system, whenever a neutron is produced, either from ssion or from a decaying precursor, another neutron is randomly chosen and eliminated (for illustration, see Fig. 4.3). Observe that the total number of precursors is left free to uctuate. We will call this scheme the N -control algorithm.

. Analysis of the moments of the total populations

We begin our analysis by addressing the behaviour of the population sizes. These equations can be readily solved, together with the initial conditions, and yield

n(t) = N m(t) = βν d,1 λ N 1 -e -λt + M e -λt .
For the N -control model, the average total precursor number m(t) exponentially relaxes to the same equilibrium solution as for the anarchic model, i.e., m ∞ = N βν d,1 /λ = N/θ. If the initial condition satises N = M θ, we simply obtain m(t) = M for any t. The equations for the second-order moments of the total population can be also derived, and read

∂ ∂t u(t) = 0 (4.24a) ∂ ∂t v(t) = βν d,1 u(t) -λ v(t) (4.24b) ∂ ∂t w(t) = βν d,1 N (2m(t) + 1) -λ(2w(t) -m(t)).
(4.24c)

The solutions to these equations are

u(t) = N 2 (4.25) v(t) = N m(t) (4.26) w(t) = m(t) 2 + m(t) -M e -2λt , (4.27)
where u, v, and w are as dened in Sec. where µ is a function of time. We make the following ansatz :

P(n, m, t) = δ nN Poiss µ(t) (m). (4.29)
Plugging this into Eq. (4.22) and simplifying leads to

∂ ∂t µ = λ(m ∞ -µ). (4.30)
Since this equation holds for any m, we can conclude that P(n, m, t) has the assumed shape provided that the ansatz also holds at t = 0 and that µ(0) = m ∞

. Derivation of the spatial master equation

In order to derive equations for the spatial moments of the neutron and precursor densities, we use again the strategy proposed in Ref. 48 : we partition the viable one-dimensional domain into a set of K equal segments of length δx = 2L/K, which we denote as V k , with k ∈ {1, . . . , K}. Correspondingly, we write n = (n 1 , . . . , n K ), m = (m 1 , . . . , m K ), and we denote by P(n, m, t) the joint probability of observing n k neutrons and m k precursors in each of the K detectors at time t, for 1 ≤ k ≤ K, given an initial condition with N neutrons and M precursors at time t 0 = 0, independently and uniformly distributed in space. The stochastic rules are those dened in Sec. 4.2 and the transition rates are specied below.

In addition to diusion, which applies only to neutrons and thus behaves the same as in a reactor without precursors, the system has the following transitions :

• A ssion in bin i and a capture in a dierent bin j (i ̸ = j), without precursor production. The corresponding transition is (a † i a j n, m) → (n, m), with rate

β 1 = β (n i + 1)(n j -1)(1 -ν d,1 ) N (N -1)
.

• A ssion and capture in the same bin but with production of one additional precursor. The corresponding transition is (n, a i m) → (n, m) with rate

β 2 = β n i (n i -1)ν d,1 N (N -1)
• A capture in bin i and a ssion in a dierent bin j (i ̸ = j) with precursor production. The corresponding transition is (a † i a j n, a j m) → (n, m) with rate

β 3 = β (n i + 1)(n j -1)ν d,1 N (N -1)
• A precursor decay in bin i and neutron capture in bin j (i ̸ = j). The corresponding transition is

(a i a † j n, a † i m) → (n, m) with rate β 4 = λ (m i + 1)(n j + 1) mN
• A precursor decay and neutron capture in the same bin. The corresponding transition is (n, a † i m) → (n, m) with rate

β 5 = λ (m i + 1)n i mN
Other transitions are possible, but they do not modify the state of the system, and as such do not appear in the master equation (for example, ssion and capture in the same bin i, without precursor production).

Probability balance then yields the master equation

∂ ∂t P(n, m, t) = i   ξ(n i+1 + 1)P(a † i+1 a i n, m, t) + ξ(n i-1 + 1)P(a † i-1 a i n, m, t) -2ξn i P(n, m, t)   + β(1 -ν d,1 )N i,j̸ =i n i + 1 N n j -1 N -1 P(a † i a j n, m, t) - n i N n j N -1 P(n, m, t) + βν d,1 N i,j̸ =i n i + 1 N n j -1 N -1 P(a † i a j n, a j m, t) - n i N n j N -1 P(n, m, t) + βν d,1 N i n i N n i -1 N -1 P(n, a i m, t) - n i N n i -1 N -1 P(n, m, t) + λm i,j̸ =i m i + 1 m n j + 1 N P(a i a † j n, a † i m, t) - m i m n j N P(n, m, t) + λm i m i + 1 m n i N P(n, a † i m, t) - m i m n i N P(n, m, t) . (4. 31
)
where ξ is the diusion rate of a neutron from a site to a neighbouring one, m = i m i is the total number of precursors, and the total number of neutrons is n = i n i = N because of population control. Once we have the space-dependent master equation, we can derive the equations for the spatial moments.

. Derivation and analysis of the spatial moments

We dene the discrete spatial moments of the neutron and precursor populations by

E[n i ](t) = n,m n i P(n, m, t) (4.32a) E[m i ](t) = n,m m i P(n, m, t), (4.32b)
for i ∈ {1, . . . , K}. The two-point correlations between particles detected in V i and particles detected in V j at time t are similarly obtained from

E[n i n j ](t) = n,m n i n j P(n, m, t) (4.33a) E[n i m j ](t) = n,m n i m j P(n, m, t) (4.33b) E[m i m j ](t) = n,m m i m j P(n, m, t), (4.33c)
for i, j ∈ {1, . . . , K}. The associated moment equations can be obtained from Eq. (4.31) by summation over all possible states, and for the average they read

∂ ∂t E[n i ](t) = ξ∆ i E[n i ](t) + λ E[m i ](t) - E[n i m](t) N (4.34a) ∂ ∂t E[m i ](t) = βν d,1 E[n i ](t) -λE[m i ](t), (4.34b)
where

∆ i f i = f i+1 -2f i + f i-1 is the discrete Laplace operator. Taking the continuous limit V i - → 0 yields ∂ ∂t n(x, t) = D∇ 2 x n(x, t) + λ m(x, t) - 1 N H n (x, t) (4.35a) ∂ ∂t m(x, t) = βν d,1 n(x, t) -λ m(x, t), (4.35b)
where we have dened the cross-moment

H n (x, t) = lim V i →0 1 V i E[n i (t)m(t)], (4.36)
with V i centered around x, and the diusion coecient

D = lim δx→0 (ξ δx 2 ). ( 4

.37)

A somewhat surprising fact is that the term H n (x, t) appearing in the equation for the average is actually a second-order moment : this means that unfortunately the hierarchy of the spatial moment equations is not closed, and analytical solutions are therefore out of reach for this model. This is to be contrasted with the case of the moments of the integral quantities, Eqs. which coincides with the solutions of the case without population control.

Similarly, the equations for the discrete pair correlation functions read

∂ ∂t E[n i n j ](t) = ξ (E[∆ i n i n j ](t) + E[∆ j n i n j ](t)) - 2β N 0 -1 E[n i n j ](t) + λC N (E[n i m j ](t) + E[m i n j ](t)) - 2λ N E[n i n j m](t) + 2β N 0 -1 δ i,j N 0 E[n i ](t) + λδ i,j E[m i ](t) + E[n i m](t) N + δ i,j ξ (E[n i+1 ](t) + E[n i-1 ](t) + 2E[n i ](t)) -δ i+1,j ξ (E[n i+1 ](t) + E[n i ](t)) -δ i-1,j ξ (E[n i ](t) + E[n i-1 ](t)) (4.40) ∂ ∂t E[n i m j ](t) = ξE[∆ i n i m j ](t) -λ N -1 N E[n i m j ](t) -λ E[n i m j m](t) N + βν d,1 C N -1 E[n i n j ](t) + λE[m i m j ](t) + δ i,j βν d,1 N N -1 E[n i ](t) -λE[m i ](t) (4.41) ∂ ∂t E[m i m j ](t) = βν d,1 (E[m i n j ](t) + E[n i m j ](t)) -2λE[m i m j ](t) + δ i,j (βν d,1 E[n i ](t) + λE[m i ](t)) .
(4.42)

When taking the continuous limit it is easier to write the equations for the modied pair correlations functions ũ(x, y, t) = u(x, t, y, t) -n(x, t)δ(x -y)

(4.43) v(x, y, t) = v(x, t, y, t) (4.44) w(x, y, t) = w(x, t, y, t) -m(x, t)δ(x -y), (4.45)
which involve third-order moments :

∂ ∂t ũ(x, y, t) = D ∇ 2 x + ∇ 2 y ũ(x, y, t) - 2β N -1 ũ(x, y, t) + λC N (v(x, y, t) + v(y, x, t)) - 2λ N H nn (x, y, t) + δ(x -y) 2βn(x, t) + 2λ N H n (x, t) (4.46a) ∂ ∂t v(x, y, t) = D∇ 2 x v(x, y, t) -λC N v(x, y, t) - λ N H nm (x, y, t) + βν d,1 C N -1 ũ(x, y, t) + λ w(x, y, t) + δ(x -y) βν d,1 N N -1 n(x, t) (4.46b)
∂ ∂t w(x, y, t) = βν d,1 (v(x, y, t) + v(y, x, t)) -2λ w(x, y, t).

(4.46c)

For the sake of conciseness, we have dened the two third-order cross-moments

H nn (x, y, t) = lim V i ,V j →0 1 V i V j E[n i (t)n j (t)m(t)] (4.47)
H nm (x, y, t) = lim

V i ,V j →0 1 V i V j E[n i (t)m j (t)m(t)].
(4.48)

and the shorthand C N = (N -1)/N . The detection volumes V i and V j are assumed to be centered around x and y, respectively. It is interesting to note that we can recognize in Eq. (4.46a) the renewal time (or, equivalently, the xation time [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF]) τ R,0 = (N -1)/(2β) for a population of prompt neutrons under population control without precursors [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. Although this denition can be extended to a population of neutrons and precursors, we obviously expect the associated renewal time to dier from τ R,0 . On a side note, Monte Carlo simulations of the N -control model for θ ≪ 1 seem to show that the relative spatial dependence of the pair correlation function is almost exactly the same as in the anarchic model for t 1 = t 2 . This is illustrated in Fig. 4.4, where we plot the shape of the correlation function for both models, for a xed value of y = -0.5 and relative to its value in x = -L = -1. This remark does not hold true for θ ∼ 1.

Note that, although the hierarchy of the moment equations is not closed, the underlying stochastic process can be straightforwardly simulated using the Monte Carlo method. In the following, we shall make extensive use of this fact to compare the N -control model, which plays the role of a reference for population control, with other models enforcing similar constraints on the population. Indeed, the occurrence of H nn (x, y, t) and H nm (x, y, t) suggests that the lack of closure originates from the fact that the N -control model correlates the size of the precursor population with the local neutron and precursor densities. It is tempting to suggest the use of the approximation

H n (x, t) ≃ n(x, t)m(t) (4.49)
in order to close Eqs. (4.35), and of similar approximations for H nn (x, y, t) and H nm (x, y, t). Doing so in the general case yields equations with time-dependent coecients in the moment equations. However, we do not know how to construct a Monte Carlo game associated to these approximate moment equations.

This eectively prevents us from validating the analytical results by comparing them with an equivalent Monte Carlo simulation, but the goodness of the approximation could be assessed by comparing the analytical results against the Monte Carlo simulations for the N -control model.

Another way to close the hierarchy of moment equations is to apply population control to both populations, so that the size of the precursor population is not a random variable. By doing so, we sacrice some physical relevance of the N -control model, but we gain the ability to derive analytical solutions. We henceforth propose two alternative models, and we discuss under which conditions they may approximate the N -control model.

. Control of neutrons and precursors

The rst alternative model consist in considering a process in which the total number of neutrons N and the total number of precursors M are kept under constraint. Conservation of N is enforced as previously described. Conservation of M is enforced as follows : when a neutron undergoes ssion and produces a precursor, another randomly chosen precursor is destroyed. When a precursor emits a delayed neutron, the precursor is not destroyed. In this way, we ensure that N and M are kept constant at any time (see Fig. 4.5). We call this algorithm the N M -control scheme. For the total population statistics, N M -control trivially yields one possible state, given by the initial condition (N, M ), with vanishing variance.

. Derivation of the spatial master equation

The spatial behaviour of the neutron and precursor populations is nonetheless non-trivial. The possible transitions (in addition to diusion) are listed below :

• A neutron capture in bin i and ssion in a dierent bin j (i ̸ = j) without precursor production.

The corresponding transition is (a † i a j n, m) → (n, m) with rate

β 1 = β (n i + 1)(n j -1)(1 -ν d,1 ) N (N -1) ,
• A neutron capture in bin i, ssion in bin j with precursor production, and destruction of another precursor in bin k, where i ̸ = j ̸ = k. The corresponding transition is (a

† i a j n, a j a † k m) → (n, m)
with rate

β 2 = β ν d,1 (n i + 1)(n j -1)(m k + 1) N (N -1)M ,
• A ssion with precursor production and a neutron capture in bin i, and precursor destruction in bin k (i ̸ = k). The corresponding transition is (n, a i a † k m) → (n, m) with rate

β 3 = β n i (n i -1)(m k + 1)ν d,1 N (N -1)M ,
• A neutron capture and precursor destruction in bin i, and a ssion with precursor production in bin j (i ̸ = j). The corresponding transition is (a

† i a j n, a † i a j m) → (n, m) with rate β 4 = β (n i + 1)(n j -1)(m i + 1)ν d,1 N (N -1)M ,
• A neutron capture in bin i, a ssion with precursor production and a precursor destruction in bin j (i ̸ = j). The corresponding transition is (a † i a j n, m) → (n, m) with rate

β 5 = β (n i + 1)(n j -1)m j ν d,1 N (N -1)M ,
• A precursor decay in bin i and a neutron capture in a dierent bin j (i ̸ = j). The corresponding transition is (a i a † j n, m) → (n, m) with rate

β 6 = λ m i (n j + 1) N M ,
By using the denitions in Section 4.2.2, we use probability balance to write the master equation

∂ ∂t P(n, m, t) = i   ξ(n i+1 + 1)P(a † i+1 a i n, m, t) + ξ(n i-1 + 1)P(a † i-1 a i n, m, t) -2ξn i P(n, m, t) + j̸ =i β(1 -ν d,1 ) N -1 (n i + 1)(n j -1)P(a † i a j n, m, t) - β(1 -ν d,1 ) N -1 n i n j P(n, m, t)   + βν d,1 (N -1)M     i,j,k i̸ =j̸ =k (n i + 1)(n j -1)(m k + 1)P(a † i a j n, a j a † k m, t) -n i n j m k P(n, m, t) i,k i̸ =k n i (n i -1)(m k + 1)P(n, a i a † k m, t) -n i (n i -1)m k P(n, m, t) i,j i̸ =j (n i + 1)(n j -1)(m i + 1)P(a † i a j n, a † i a j m, t) -n i n j m i P(n, m, t) i,j i̸ =j (n i + 1)(n j -1)m j P(a † i a j n, m, t) -n i n j m j P(n, m, t)     + λ N i,j i̸ =j m i (n j + 1)P(a i a † j n, m, t) -m i n j P(n, m, t) , (4.50)
where n = N and m = M at all times.

. Derivation and analysis of the spatial moments

By multiplying by n i (resp m i ) the neutron (resp precursor) population in spatial cell i and summation over all possible states, the equations for the rst spatial moments are given by

∂ ∂t E[n i ](t) = ξ∆ i E[n i ](t) + λ E[m i ](t) -E[n i ](t) M N (4.51) ∂ ∂t E[m i ](t) = βν d,1 E[n i ](t) -E[m i ](t) N M , (4.52)
showing that the constraint of xed (N, M ) indeed sidesteps the closure issues of the N -control model.

The equations for the rst-order moments are now closed and, taking the continuous limit, they read :

∂ ∂t n(x, t) = D∇ 2 x n(x, t) + λ m(x, t) -n(x, t) M N (4.53a) ∂ ∂t m(x, t) = βν d,1 n(x, t) -m(x, t) N M . ( 4

.53b)

If N and M are in the equilibrium ratio N/M = θ = λ/(βν d,1 ), then Eqs. (4.53) reduce to the usual forward equations for the rst moments of freely evolving populations of neutrons and precursors [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF].

This has two consequences. First, we can conclude that the mixing time for the N M -control model is the same as for the anarchic model, τ 1 . Second, Eqs. (4.53) admit uniform and constant asymptotic solutions

n(x, t) → n ∞ (x) = N 2L (4.54a) m(x, t) → m ∞ (x) = M 2L . ( 4

.54b)

For the second-order moments we obtain once again a closed system of equations, rst in its discrete formulation :

∂ ∂t E[n i n j ](t) = ξ (E[∆ i n i n j ](t) + E[∆ j n i n j ](t)) - 2β N -1 + 2λM N E[n i n j ](t) + λC N (E[m i n j ](t) + E[n i m j ](t)) + 2N β N -1 δ i,j E[n i ](t) + λδ i,j E[m i ](t) + E[n i ](t) M N + δ i,j ξ (E[n i+1 ](t) + E[n i-1 ](t) + 2E[n i ](t)) -δ i+1,j ξ (E[n i+1 ](t) + E[n i ](t)) -δ i-1,j ξ (E[n i ](t) + E[n i-1 ](t)) (4.55) ∂ ∂t E[n i m j ](t) = ξE[∆ i n i m j ](t) - βν d,1 N M + λ M N E[n i m j ](t) + βν d,1 N -2 N -1 E[n i n j ](t) + λE[m i m j ](t) + βν d,1 N (N -1) δ i,j E[n i ](t) (4.56) ∂ ∂t E[m i m j ](t) = βν d,1 C M (E[m i n j ](t) + E[n i m j ](t)) (4.57) - 2βν d,1 N M E[m i m j ](t) + βν d,1 δ i,j E[n i ](t) + E[m i ](t)
N M

.

Taking the continuous limit, we have then :

∂ ∂t ũ(x, y, t) = D ∇ 2 x + ∇ 2 y ũ(x, y, t) -τ -1 n ũ(x, y, t) + λC N (v(x, y, t) + v(y, x, t)) + 2βδ(x -y)n(x, t) (4.58a) ∂ ∂t v(x, y, t) = D∇ 2 x v(x, y, t) -τ -1 c v(x, y, t) + βν d,1 C N -1 ũ(x, y, t) + λ w(x, y, t) + δ(x -y) (2βν d,1 n(x, t) + λm(x, t)) (4.58b) ∂ ∂t w(x, y, t) = βν d,1 C M (v(x, y, t) + v(y, x, t)) -τ -1 p w(x, y, t), (4.58c)
with C M = (M -1)/M and where we dened the time constants

τ n = 2β N -1 + 2λM N -1 (4.59a) τ c = βν d,1 N M + λ M N -1 (4.59b) τ p = M 2βν d,1 N , (4.59c)
Equations (4.58) are a linear system of equations for (ũ, v, w). Following the heuristic arguments of Zhang et al. [START_REF] Zhang | Diusion reproduction processes[END_REF], we should expect the renewal time for our system to be determined by the dominant time constant of the correlation functions. In the N M -control model, the collective modes of the correlation functions are associated to the eigenvalues of the matrix of the coecients of (ũ, v, w) in Eqs. (4.58),

which reads R =   -τ -1 n 2λC N 0 βν d,1 C N -1 -τ -1 c λ 0 2βν d,1 C M -τ -1 p   . (4.60)
Assuming that N and M are chosen in the equilibrium ratio N = θM and that N is large, we can extract the scaling behaviour of the eigenvalues of R with respect to θ, for small θ. (4.62)

r 3 ≃ - 2βθ 2 (1 + 3ν d,1 ) N (1 + θ) 2 , (4.63)
with r 1 < r 2 < r 3 . The time constant associated to the dominant eigenvalue, r 3 , is the renewal time for the N M -control model, and reads

τ N M R ≃ N (1 + θ) 2 2βθ 2 (1 + 3ν d,1 ) . (4.

64)

This expression is very similar to the extinction time for the anarchic scheme, Eq. (3.57). However, the renewal time τ N M R is not related to the extinction time for the system, because no critical catastrophe is possible in the N M -control model.

Similarly to how we did for the anarchic scheme, we can then dene a dimensionless clustering parameter η N M = τ 1 /τ N M R . However, given that the mixing time of the N M -control scheme is the same as the mixing time of the anarchic scheme, and that the renewal time τ N M R is very similar to the extinction time of the anarchic scheme, the clustering parameter also results in a 1/θ scaling for small θ, and the remarks made about the anarchic scheme apply verbatim to the N M -control scheme.

The full solution for Eqs. (4.69)

The linear system of equations solved by the Fourier coecients can be obtained from Eqs. Indeed, in this case all the coecients with k x ̸ = k y are zero. The terms f k,k thus read

f ∞ = +∞ k=-∞ f k,k cos kπ 2L (L -x) cos kπ 2L (L -y) .
ũk,k = m(N -1)λ 2 + nβ N (α 1 + k2 D) + 2(N -1)ν d,1 λ LN α 1 + k2 D τ n + 2 k2 D -2L(N -2)βν d,1 λ (4.71) v k,k = 2nN β (N -1)τ n + (N -2)β + 2 k2 (N -1)D ν d,1 + m(N -1)N τ n + 2 k2 D λ 2L(N -1) N (α 1 + k2 D)(τ n + 2 k2 D -2(N -2)βν d,1 λ) (4.72) wk,k = M -1 N ṽk,-k (4.73)
where τ n , τ c are given by Eqs. For θ = 0.001 (Fig. 4.6a), the N M -control scheme provides a good approximation of the N -control scheme, and all the curves representing neutron correlations are very similar. This means that, when θ ≪ 1, which is the case in nuclear reactors, a system where neutrons and precursors are both kept under control has similar correlation functions as a system where population control acts on neutrons alone.

On the other hand, when θ = 1 (Fig. 4.6b), the N -control and N M -control models yield similar, but dierent correlation functions. Indeed, we observe that the N M -control scheme results in stronger spatial short-range correlations compared to the N -control scheme, while long-range correlations are weaker in the N M -control scheme. This can be easily explained : in the N -control scheme, when a precursor decays, it is destroyed and replaced by a neutron that immediately starts diusing. This mechanism can be seen as a kind of delayed diusion ; when a neutron induces a ssion event and produces a precursor, it eectively suspends diusion for a time of the order of 1/λ. On the other hand, in the N M -control model, a precursor does not die when it decays, and thus plays the role of a source at a xed position until it is replaced by another precursor. The precursor density can then undergo signicant local uctuations favouring short-range correlations by inducing overproduction of neutrons in a small volume, at the expense of long-range correlations.

Some insight about the similarity between the N -control and N M -control schemes for θ ≪ 1 can be gained by comparing Eqs. (4.46) and Eqs. (4.58). Indeed, remember that θ ≪ 1 means that the number of precursors is much larger than the number of neutrons. If m(t) is suciently large, which is the case here, its uctuations will be negligible compared to its mean value. This allows us to consider m(t) as a deterministic observable. If we then assume that

E[n i (t)n j (t)m(t)] ≃ E[n i (t)n j (t)]E[m(t)],
then Eq. (4.46a) reduces to Eq. (4.58a). Thus, in this limit, neutron-neutron correlations follow the same dynamics in both schemes. It is worth remarking however that, under the same approximation, the equations for v and w in the N -control model do not reduce to the corresponding equations in the N M -control model. Similarly to the prompt system, we can complement the analysis of spatial moments by investigating the statistical behaviour of neutron families.

. Analysis of the statistics of neutron families

The analysis of the time and space correlations of the N M -control can be usefully complemented by the investigation of the statistics of neutron families, which relies on a generalization of the approach discussed in Sec. 4.1.3 for the simpler case of the constrained model without precursors.

For the N M -control model, both source neutrons and precursors are assigned an identier (an integer in {1, . . . , F }), which they transmit to their descendants. Let n i (m i ) be the number of neutrons (precursors) with identier i. Particles sharing the same identier form a family, so that the simulation starts with F initial families. The size of a family increases by one when one of its members induces ssion without precursor production, and by two if it is a ssion with precursor production. The size decreases by one when a neutron or precursor is killed.

To the best of our knowledge, the eect of introducing delayed events in a time-dependent Galton-Watson process has not been investigated in the context of family xation. We focus on the case where the number of neutrons and precursors satises the ratio M = N/θ, with θ = λ/βν d , which corresponds to the equilibrium initial condition.

To write the recurrence equations describing the conditional moments of the xation time τ we need to consider all the `useful' transitions, i.e. those that change the number of neutrons or precursors in family A, and the corresponding rates and probabilities. There are 6 such transitions that can be deduced from Section 4.3 :

• (n, m) → (n + 1, m) with rate β (+1,0) = β(N -n)n N -1 (1 -ν d ) + ν d m M + λ N m(N -n). • (n, m) → (n -1, m) with rate β (-1,0) = β(N -n)n N -1 (1 -ν d ) + ν d (M -m) M + λ N n(M -m). • (n, m) → (n, m + 1) with rate β (0,+1) = βν d (N -1)M n(n -1)(M -m). • (n, m) → (n, m -1) with rate β (0,-1) = βν d (N -1)M (N -n)(N -n -1)m. • (n, m) → (n + 1, m + 1) with rate β (+1,+1) = βν d (N -1)M n(N -n)(M -m). • (n, m) → (n -1, m -1) with rate β (-1,-1) = βν d (N -1)M (N -n)nm.
We also dene the rate of useful transitions

β u (n, m) = v∈T β v (n, m),
where T = {(±1, 0), (0, ±1), ±(1, 1)}. Observe that β u ≤ β t , where β t = βN + λM represents the total transition rate, since not all transitions contribute to xation. Subsequently, one can dene the probability that a transition v happens, conditioned to the fact that the transition is useful : p v (n, m) = β v /β t (n, m).

One might equivalently consider all transitions (instead of only useful transitions), in which case β u and β t coincide, and the transition (n, m) → (n, m) should be explicitly taken into account.

Fixation probability

In this situation, we do not know a priori the xation probability f (n, m). However, from the law of total expectation we write

f (n, m) = v∈T p v (n, m)f (n + v 1 , m + v 2 ), (4.74)
with boundary conditions f (N, M ) = 1 and f (0, 0) = 0. Equation (4.74) admits the solution

f (n, m) = N n + M θm N 2 + θM 2 .
(4.75)

For M = N/θ, Eq. (4.75) reduces to (n + m)/(N + M ) : regardless of their nature, all individuals have the same probability of fathering the xating family. Moreover, the linearity of the xation probability f suggests that neutrons and precursors behave like two independent populations with respect to xation.

The solution in Eq. (4.75) has been carefully veried against Monte Carlo simulations (not shown here for conciseness). Knowledge of f will now allow us to apply our methodology to the problem of the xation time.

Moments of the xation time

We investigate the statistical moments of the xation time. Let us rst consider the average (conditional) xation time E[τ |A](n, m). Using the law of total expectation and dening an intermediary function

g 1 (n, m) = f (n, m)E[τ |A](n, m) for (n, m) ̸ = (0, 0) and g 1 (0, 0) = 0, we obtain v∈T p v (n, m)g 1 (n + v 1 , m + v 2 ) -g 1 (n, m) = - v∈T p v (n, m)f (n + v 1 , m + v 2 )
β t (n, m) .

(4.76)

The presence of non-constant coecients and the shape of the source term make Eq. (4.76) harder to solve analytically. However, Eq. (4.76) is formally a multi-dimensional dierence equation ; it can be thus solved numerically by regarding it as a system of non-homogeneous linear equations for the (N + 1)(M + 1) unknowns g(n, m), with (0 ≤ n ≤ N , 0 ≤ m ≤ M ). The equations provided by Eq. (4.76) are complemented with equations stemming from boundary conditions imposing that g 1 (n, m) = 0 outside of the domain. The linear system is then solved numerically, and results are successfully compared to Monte Carlo simulations in Fig. 4.7.

The variance of the xation time can be obtained from the second moment, which, after using the law of total expectation and dening an intermediate function g We conclude by revisiting the relation between family statistics and spatial clustering. Again, due to the choice of mass-preserving boundary conditions, the statistics of families is insensitive to spatial eects. Due to the numerical nature of our solution, we can only provide empirical scaling relations for the

2 (n, m) = f (n, m)E[τ 2 ](n, m) for (n, m) ̸ = (0, 0) and E[τ 2 ](0, 0) = 0, is determined by v∈T p v (n, m)g 2 (n + v 1 , m + v 2 ) -g 2 (n, m) = v∈T 2p v (n, m)g 1 (n + v 1 , m + v 2 ) β u (n, m) + p v (n, m)f (n + v 1 , m + v 2 ) β u (n, m) 2 .
xation time τ 1 = E[1, 0] (or equivalently E[0, 1]). Assuming N ≫ 1, we can show that τ 1 ∼ N/βν d θ 2
when θ ≪ 1 (numerical investigations not detailed here for conciseness) ; conversely, τ 1 ∼ N/βν d when θ ≫ 1). Given that the equilibrium ratio M = N/θ is enforced, M does not appear explicitly in this expression. For the case of particles evolving in unbounded homogeneous systems, the time evolution of ⟨r 2 (t)⟩ is characterized by a single timescale, i.e, the renewal time τ R , whose leading-order term reads

τ R ∼ N (1 + θ) 2 2βθ 2 (1 + 3ν d ) , (4.78

. Immigration model

We have observed that the precursor population exhibits only weak uctuations in the θ ≪ 1 limit.

This suggests another alternative model where precursors are modeled as a xed external source, with the aim of obtaining a time-dependent solution that, in this limit, successfully approximates the Ncontrol model. Thus, we model the system as a collection of neutrons under population control, with an external, time-independent source term modelling precursor decay. Formally, this model is equivalent to an immigration model with a time-independent, Poissonian immigration source describing the asymptotic precursor decay density, given by

Q I (x) = λm ∞ (x) = λ θ n ∞ (x), (4.79)
where θ is the parameter of the equivalent N M -control scheme. Population control is enforced by requiring that each time a new neutron enters the system, whether by ssion or from the external source, we destroy another randomly chosen neutron, as illustrated in Fig. 4.9. Neutrons diffuse (in red) until they branch or are killed by another neutron branching somewhere else (vertical blue dashed lines show these correlated events). They can also be produced by the source, thus killing another randomly chosen neutron.

. Derivation of the spatial master equation

In addition to diusion, the transitions involved in this model are

• A neutron capture in bin i and a ssion in bin j (i ̸ = j). The corresponding transition is a † i a j n → n with rate

β 1 = β(n i + 1)(n j -1) N (N -1) ,
• A neutron capture in bin i and an emission by the Poisson source in a dierent bin j (i ̸ = j). The corresponding transition is a † i a j n → n with rate

β 2 = λM Q j (n i + 1) N ,
where Q j is the probability that the emission happens in bin j.

Following the usual derivation method, we obtain rst the master equation associated to this stochastic process

∂ ∂t P(n, t) = i   ξ(n i+1 + 1)P(a † i+1 a i n, t) + ξ(n i-1 + 1)P(a † i-1 a i n, t) -2ξn i P(n, t) + β N -1 j̸ =i (n i + 1)(n j -1)P(a † i a j n, t) -n i n j P(n, t) + λM N j̸ =i Q j (n i + 1)P(a † i a j n, t) -n i P(n, t)   .
(4.80)

. Derivation and analysis of spatial moments

The equation for the discrete average neutron density is again obtained by summation from Eq. (4.80),

namely ∂ ∂t E[n i ](t) = ξ∆E[n i ](t) - λM N E[n i ](t) + λM Q i , (4.81)
which, by taking the continuous limit, becomes

∂ ∂t n(x, t) = D∇ 2 n(x, t) - λ θ n(x, t) + Q I (x).
(4.82)

Taking Q I (x) as given by Eq. (4.79), the last two terms asymptotically cancel out and the dynamics of the average is driven only by diusion, similarly to what happens in the anarchic model when the system is critical.

The equation for the discrete pair correlation function reads

∂ ∂t E[n i n j ](t) = ξ (E[n i ∆n j ](t) + E[∆n i n j ](t)) - 2β N -1 + 2λM N E[n i n j ](t) + δ i,j 2N β N -1 E[n i ](t) + λM δ i,j Q i + E[n i ](t) N + λM N -1 N (Q j E[n i ](t) + Q i E[n j ](t)) + δ i,j ξ (E[n i+1 ](t) + E[n i-1 ](t) + 2E[n i ](t)) -δ i+1,j ξ (E[n i+1 ](t) + E[n i ](t)) -δ i-1,j ξ (E[n i ](t) + E[n i-1 ](t)) . (4.83)
which, under the continuous limit and after dening ũ(x, y, t) = u(x, y, t) -δ(x -y)n(x, t), becomes for the average neutron density, and for the pair correlation function, where G ′ is the Green's function associated to Eq. (4.82). As a side note, taking the λ → 0 limit in this equation we recover the pair correlation function for a system of prompt neutrons under population control [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF][START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF].

∂ ∂t ũ(x, y, t) = D ∇ 2 x + ∇ 2 y ũ(x, y, t) -τ -1 n ũ(x, y, t) + C N (n(x, t)Q I (y) + n(y, t)Q I (x)) + 2βδ(x -y)n(x,
ũ(x, y, t) = N (N -1) 4L 2 exp - t τ n + 2τ n λM (N -1) 4L 2 1 -exp - t τ n + 2βN 2L dx ′ t 0 dt ′ G ′ (x, t|x ′ , t ′ )G ′ (y, t|x ′ , t ′ ) exp - t -t ′ τ n
Comparison with Monte Carlo simulations (see Fig.

. Mean squared pair distance functions

We now discuss the mean-squared pair distance for our population control models, and we compare it to the case of an anarchic population.

For the N -control scheme, we do not have access to an analytical expression for the mean-squared pair distance, because the moment equations, Eqs. (4.46), are not closed. However, we do have an asymptotic correlation function for the N M -control scheme, given by Eqs. (4.73), from which we can deduce an asymptotic mean-squared pair distance. For reection boundary conditions, it is given by

⟨r 2 N M ⟩ = 8L 4 ũ0,0 3N 2 - 16L 2 π 2 N 2 +∞ k=1 ũk,k k 4 , (4.87)
where ũk,k is the Fourier coecient for the neutron-neutron correlations asymptotic solution of Eq. (4.58a).

On physical grounds, we expect the asymptotic pair distance in the N M -control scheme to be always smaller than the uncorrelated value, Eq. (3.61).

As for the immigration model, an analytical expression for the mean-squared pair distance can be obtained by applying the denition of the mean-squared pair distance function, Eq. (3.10), to the solution, Eq. (4.86). For reection boundary conditions, straightforward calculations hence yield The sum can be computed analytically in the asymptotic limit and reads

⟨r 2 I ⟩(t) = C N 2L 2 3 - 128L 2 β N +∞ k=1 1 -e -(τ -1 n -2α k )t (kπ) 4 τ -1 n -2α k
⟨r 2 I ⟩ ∞ = C N 2L 2 3 + 16βτ n N × Dτ n L 2 - 2Dτ n 3 - 4L 2 45 - √ 2(Dτ n ) 3 2 L cot 2τ n D L (4.89)
If we take the mean-squared pair distance as an indicator of clustering, Fig. 4.11a clearly illustrates that the three models have very similar clustering behaviour for θ ≪ 1, even for very long times. For θ ∼ 1, on the other hand, the models behave dierently, as shown in Fig. 4.11b. In particular, we note that the immigration model yields the largest mean-squared pair distance of all the considered control schemes. This is easily explained by the fact that the `decay source' of the immigration model is assumed to be uniformly distributed in space and completely uncorrelated with the neutrons, an assumption that is bound to reduce clustering. The N M -control scheme yields a smaller mean-squared pair distance than the N -control scheme, indicating that clustering is more prominent in the former. This behavior can be explained by observing that a precursor may in fact produce several neutrons at the same position before being destroyed, which tends to increase clustering. It is interesting to note that for all control models the squared pair distance asymptotically converges to ⟨r 2 ⟩ ∞ < ⟨r 2 ⟩ iid , because the critical catastrophe is avoided, contrary to what happens in the anarchic case. However, Eqs. (4.87) and (4.89) show that ⟨r 2 ⟩ ∞ tends to the uncorrelated value as N tends to innity.

. Conclusions and perspectives

In an attempt to take into account population control eects, we introduced several idealized sampling strategies. These models break the statistical independence of the neutron ssion chains, which forced us to abandon the Pál and Bell backward formalism in favour of the forward formalism. We derived the master equations for four dierent population-control models :

• in the prompt model without precursors, the neutron population is kept constant. We provided a formal derivation for the spatial moment equations instead of the previous derivation relying on heuristic arguments ;

• in the N -control model, the neutron population is kept constant, but no constraint is applied to the precursor population. This model yields a non-closed hierarchy of moment equations ;

• in the N M -control model, both the neutron and precursor populations are controlled. This model yields a closed hierarchy of moment equations. We identied the dominant timescales of the model and presented asymptotic solutions for the rst two spatial moments.

• in the immigration model, precursors are modelled as a uniform Poisson neutron source whose intensity is equal to the asymptotic precursor decay rate at equilibrium.

For each model, we obtained two-point, one-time pair correlation functions by analytical means when possible, and compared them to analog kinetic Monte Carlo simulations. Such comparisons showed that all the models including precursors yield very close results when θ ≪ 1, which is the main regime of interest for nuclear reactors. We also showed that the introduction of population control prevents the occurrence of the critical catastrophe and thus quenches clustering.

Within the same theoretical framework, we also investigated family xation, i.e. the eventual survival of a single particle family. In the case where precursors are neglected, we derived and analytically solved the equations for the rst two moments of the xation time starting with any number of neutrons. In particular, when there is a single initial neutron in each family, we showed that the average xation time is given by E[τ p |A](1) = (N -1) 2 /βN . Furthermore, we discussed the relation between xation and clustering. For the case of unbounded media, ⟨r 2 (t)⟩ has an exponential saturation with a typical timescale τ R,p = N/2β, which suggests that clustering and xation are intimately related, their respective timescales being similar when N ≫ 1.

In the N M -control model, which includes precursors, we derived the equations for the rst two moments of the xation time, and we solved them numerically. We determined that, in this case, the average xation time scales as τ 1 ∼ N/βν d θ 2 when θ = N/M ≪ 1. The solutions were veried against Monte Carlo simulations. Similarly as in the prompt case, we found that for the case of unbounded media the typical saturation time of the square pair distance ⟨r 2 (t)⟩ has almost the same scaling as the average xation time τ 1 when N ≫ 1.

In Part I we investigated the behaviour of space-time correlations in analog Monte Carlo simulations of diusion-based stochastic models including neutrons and precursors. We developed an analytical framework for the characterization of space-time correlations by making several assumptions : the energy dependency was neglected ; particle displacements were replaced with Brownian Motion ; the system was assumed to be spatially homogeneous with mass-preserving boundary conditions ; only the following events were considered : ssion, capture, and precursor decay. This model can be treated analytically, at the expense of being considerably simplied.

The results obtained in Part I provide a clear setting for the investigation of kinetic Monte Carlo simulations, but are by construction restricted to the case of analog sampling, without any variance-reduction method, and only apply to single-speed diusion in spatially homogeneous systems. To overcome these imitations, and broaden the domain of validity of our ndings, in Part II, we will relax many of the assumptions we made so far. In particular, while some simplications will be retained for the ease of computation (use of multi-group formalism, with a set of collision events including only ssion, sterile capture, isotropic scattering and precursor decay), we will now move to fully non-analog Monte Carlo simulations, covering a large spectrum of variance-reduction and population-control strategies. Our main goal for Part II will be to assess the impact of such techniques on the behaviour of the uctuations and correlations in kinetic simulations.

Deuxième partie

Correlations in time-dependent transport problems solved by non-analog Monte Carlo methods

-Spatial and generational correlations in Monte Carlo eigenvalue calculations

As shown in Chapter 2, the critical source for kinetic problems depends on the fundamental eigenmode of the k-eigenvalue formulation of the Boltzmann equation. Within the framework of Monte Carlo simulation, the fundamental eigenmode and the associated dominant eigenvalue can be obtained using the well-known power iteration method, which relies on the iteration of the ssion source : neutrons starting a given `generation' are chosen amongst the neutrons born from ssion in the previous generation [2,[START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. After a sucient number of generations, the phase-space distribution of ssion neutrons eventually becomes stationary, and so do the related tallies, such as neutron ux and reaction rates.

Once the ssion source has converged to a stationary state, the sought tallies are averaged using an ergodic average over successive generations in order to reduce the statistical uncertainty on the estimated values. For the purpose of estimating condence intervals on the ergodic averages, for the sake of simplicity most production-level Monte Carlo codes ignore correlations between successive generations, although some statistical tests are becoming progressively available [START_REF] Brown | Diagnostics for undersampling and clustering in Monte Carlo criticality calculations[END_REF], fostered by the increased awareness of Monte Carlo practitioners. Ground-breaking investigations concerning the impact of correlations on the estimation of variance in Monte Carlo power iteration have been carried out by Brissenden and Garlick, who noted that inference techniques based on an assumption of independent and identically distributed samples led to an underestimation of the sample variance of the average multiplication factor k e [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF]. The severity of the underestimation was later related to the value of the dominance ratio (i.e.

the ratio of the second over the rst eigenvalue of the Boltzmann equation) by Ueki et al. [START_REF] Ueki | Autocorrelation and dominance ratio in Monte Carlo criticality calculations[END_REF], and since then several authors have tried to predict and correct the estimates of the variance [START_REF] Herman | Monte Carlo and thermal hydraulic coupling using low-order nonlinear diusion acceleration[END_REF]3437].

More recently, Dumonteil et al. have pointed out that ssion-induced correlations may lead to a strong spatial patchiness (i.e. non-Poisson spatial uctuations) in the ssion source, which was dubbed neutron clustering [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF]. A characterization of spatial correlations relying on moment equations was derived in a time-dependent context in simplied nuclear reactor models [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF][START_REF] Bonnet | Space and time correlations for diusion models with prompt and delayed birth-and-death events[END_REF], and the key ndings were shown to apply also to power iteration [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF][START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF] and to production-level Monte Carlo simulations of realistic congurations [START_REF] Dumonteil | Patchy nuclear chain reactions[END_REF]. Carlo simulations, in that the estimate of the variance is asymptotically not altered [START_REF] Mickus | Does neutron clustering aect tally errors in Monte Carlo criticality calculations?[END_REF].

Given the lack of consensus on these matters, which are highly relevant for both conceptual and practical questions related to power iteration, we have decided to explore in more depth the emergence of spatial correlations and how they are aected by the key mechanisms at play in Monte Carlo simulations, namely the collision sampling strategies and the population-control and variance-reduction methods. Since the statistical behaviour of the Monte Carlo estimate of the fundamental mode has a direct impact on the sampling of the critical source for kinetic problems, the quantication of the correlations aecting particle transport in the course of the power iteration algorithm is of utmost importance for the investigations carried out in this thesis work. In the following we will therefore briey recall the main ideas behind the Monte Carlo power iteration, provide a thorough examination of the interplay between generational and spatial correlations, and show how they are aected by several variance-reduction and population-control techniques. We will focus in particular on the use of branchless collisions, which has been independently introduced in the context of power iteration calculations by Fröhlicher et

. Power iteration in Monte Carlo simulation

As recalled in Sec. 2.1.4, the k-eigenvalue formulation of the Boltzmann equation is :

Ω • ∇φ k (r, Ω, E) + Σ t (r, E)φ k (r, Ω, E) = Σ s (r, Ω ′ → Ω, E ′ → E)φ k (r, Ω ′ , E ′ )dΩ ′ dE ′ + 1 k χ p (r, E) 4π 4π ν p (r, E ′ )Σ f (r, E ′ )φ k (r, Ω ′ , E ′ ) dΩ ′ dE ′ , (5.1)
+ 1 k j χ j d (r, E) 4π ν j d (r, E ′ )Σ f (r, E ′ )φ k (r, Ω ′ , E ′ )dΩ ′ dE ′ (5.2)
where k is the eigenvalue and φ k the corresponding eigenfunction. Notation is the same as in Sec. 2.1.4. The dominant eigenpair {k 0 , φ k 0 } of Eq. (5.2) can be estimated using a stochastic equivalent of the power iteration method, as described in Sec. 2.1.4. In the following, we shall present the specic aspects of our implementation.

. Variance reduction and population control methods

In order to prevent the population size from growing out of control or dying out over the course of many generations, the total statistical weight of the ssion source is kept constant. Fixing the total statistical weight of the source is also crucial for normalizing the scores to one ssion neutron. The statistical dispersion of neutron weights can be further reduced by using optional population control algorithms : i) local population control may be applied in the course of a generation on individual neutrons (e.g. Russian roulette and splitting are applied to neutrons emerging from collision events [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]), and ii) global population control can be applied to the ssion bank, before neutrons are promoted to source neutrons for the following generation (e.g. weight combing is applied to ssion neutrons in order to select source neutrons [START_REF] Booth | A weight (charge) conserving importance-weighted comb for Monte Carlo[END_REF]).

The sampling of neutron histories can be either analog or non-analog ; in the latter case, neutrons are assigned weight correction factors to keep the Monte Carlo game unbiased. Here, the sampling of the ight kernel will always be analog, whereas for collisions we will use various non-analog sampling methods, in order to probe their eectiveness as variance-reduction techniques. We shall use implicit capture and forced ssion (see Sec. 2.2.3), which are customary in most production Monte Carlo codes [START_REF] Brun | TRIPOLI-4®, CEA, EDF and AREVA reference Monte Carlo code[END_REF]9395].

For convenience, we remind the formula to sample the number of ssion neutrons :

n f = ξ + ν(r, E)Σ f (r, E) Σ t (r, E) (5.3)
where ξ ∼ U[0, 1] and ⌊•⌋ denotes the integer part. This formula is used twice, to sample the number of prompt neutrons (with ν = ν p ), and the number of delayed neutrons (with ν = ν d ) ; the family of each delayed neutron is then sampled in an analog manner, with probabilities ν j d /ν d . The energy of the ssion neutrons are sampled from the corresponding spectra (χ p or χ j d ) ; the direction is sampled isotropically. Fission neutrons are assigned the weight w of their parent, regardless of their nature. Finally, the parent neutron is forced to undergo scattering, and the eect of absorption is taken into account by correcting the statistical weight by the survival probability, namely, ,E) .

w ′ = w Σ s (r, E) Σ t (r
(5.4)

In this non-analog scheme, particles never disappear by absorption ; histories can be terminated by either leakage or Russian roulette after collision events [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF].

The sampling strategy described so far belongs to the class of `branching' algorithms, in the sense that new particles (hence new branches of the particle histories) can be created at collision events through the sampling of ssion progeny. In what follows, we shall also use branchless collisions, that ensure that exactly one single neutron is sampled at collisions [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. The idea and basic implementation of branchless collisions are exposed in Sec 2.2.3. Here, we shall recall the expression for the weight modier :

w ′ = w Σ s + νΣ f Σ t .
(5.5)

The probabilities to select scattering or ssion then read

P s = Σ s νΣ f + Σ s and P f = νΣ f νΣ f + Σ s , (5.6)
respectively. The use of branchless collisions in power iteration calculation was proposed independently by us in Ref. [START_REF] Belanger | The eect of branchless collisions on neutron clustering[END_REF] and by Frohlicher et al. in Ref. [START_REF] Fröhlicher | Generational variance reduction in monte carlo criticality simulations as a way of mitigating unwanted correlations[END_REF], and requires some adjustments compared to its usual application in kinetic simulations : if scattering is chosen, energy and direction are sampled as customary, and the emitted neutron performs a new ight within the same history. Otherwise, if ssion is chosen, the emitted neutron is sent to the ssion bank and the current history is terminated : each neutron undergoes at most one ssion event during a generation, producing at most one descendant per generation 1 .

With the branchless sampling strategy, population control is mandatory : the number of neutrons would otherwise decrease over generations, since neutrons disappearing by leakage or Russian roulette would not be replaced. In practice, applying splitting to particles emerging from collision events suces to counteract this limitation, provided that the number of particles per generation is suciently large.

Note that splitting may reintroduce correlations between neutrons, which suggests that a trade-o must be found between keeping the population size large enough, and reducing the number of splitting events.

In this chapter, we will compare the performance of branchless collisions with a method proposed in Ref. [START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF], which relies on a last-event estimator for the ssion source [START_REF] Spanier | Monte Carlo principles and neutron transport problems[END_REF]. The sampling of the collision channel is kept analog (i.e. the statistical weight w of a given neutron is constant throughout a generation), although the statistical weight may vary from one neutron to another, and from one generation to the next generation. Upon sampling absorption (capture or ssion), the neutron history is terminated, and an average weight of w f ssion neutrons is sent to the ssion bank, where E) .

w f = w ν( - → r , E)Σ f ( - → r , E) Σ a ( - → r ,
(5.7)

We chose to implement an unweighted strategy for populating the ssion bank : n f = ⌊ξ + w f ⌋ neutrons of unit statistical weight are sent to the ssion bank ("U method" ; here ξ ∼ U([0, 1])) [START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF]. This method for populating the ssion bank may be also applied to branching and branchless collisions, with an adequate choice of w f : for branchless collisions, we simply have w f = w, while for branching collisions we have E) .

w f = w ν( - → r , E)Σ f ( - → r , E) Σ t ( - → r ,
(5.8)

Other population control mechanisms can be applied to power iteration. One such method is combing, originally proposed by Booth [START_REF] Booth | A weight (charge) conserving importance-weighted comb for Monte Carlo[END_REF] and recently explored at length in the context of time-dependent simulations [START_REF] Variansyah | Analysis of population control techniques for time-dependent and eigenvalue Monte Carlo neutron transport calculations[END_REF][START_REF] Faucher | New kinetic simulation capabilities for Tripoli-4®: Methods and applications[END_REF] ; one can also refer to Section 2.2.3 above. As a reminder, the idea behind combing is essentially to enforce a constant number of particles in the source bank of each generation, while at the same time preserving the total weight. The source particles are sampled using a constant weight interval over the total weight, starting with a random oset that guarantees unbiasedness.

An alternative to combing relies on the sampling without replacement (WOR) method, proposed by Sutton [START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF], which is an improved version of the Duplicate-Discard strategy presented by Variansyah et McLarren [START_REF] Variansyah | Analysis of population control techniques for time-dependent and eigenvalue Monte Carlo neutron transport calculations[END_REF]. In this method, N source neutrons for the next generation are sampled without replacement from the list of the N ′ ssion sites that stem from the previous generation. In addition, when the size of the ssion bank is smaller than the sought ssion source size (i.e. N ′ < N ), Sutton suggests a two-step procedure : rst, each neutron in the ssion bank is copied ⌊N/N ′ ⌋ times (with statistical weight), and the remaining N -N ′ ⌊N/N ′ ⌋ are then sampled without replacement. Thus, although this method is "without replacement", the same ssion site can nonetheless be selected more than once when N ′ < N . This ensures that the number of neutrons at the beginning of each generation is constant.

The implementation of sampling WOR described in Ref. 42 applies to a simplied quasi-analog model where all particles share the same statistical weight, and the generalization to a more realistic framework including non-analog particle transport is straightforward : neutrons are sampled uniformly and without replacement from the list of (unweighted) ssion sites, and are assigned the statistical weight of the ssion site they originate from, which in our case is one. Finally, using the U method always leads to the same total weight, and all source neutrons will have a unit weight.

. Estimating the correlations of the sought observables

Most production simulation codes perform ergodic averages of the sought observables over successive `active' generations, after convergence has been attained, and compute an apparent variance assuming (for ease of computation) that successive generations are independent. In practice, generational correlations cause the apparent variance to underestimate the real variance of the stochastic process. Several methods have been developed to recover the real variance from the apparent variance, for instance by block-averaging tallies over several generations in order to wash out correlations, or by modeling generational correlations in the system and applying suitable corrections to the apparent variance [3436,[START_REF] Demaret | Accurate determination of condence intervals in Monte Carlo eigenvalue calculations[END_REF].

Alternatively, one might take ensemble averages over fully independent power iteration replicas : in this case, the real variance of the stochastic process would be estimated, but this would come at the cost of an often intolerable increase in simulation time.

By construction, the collision sampling strategies and the variance-reduction and population-control methods described in Sec. 5.1.1 do not aect the estimate of the average, since these algorithms are precisely chosen to ensure a fair (i.e. unbiased) Monte Carlo game. Although Fröhlicher et al. suggested that spatial clustering may induce a bias in the estimation of the neutron ux, the number of inactive generations they had chosen for their system appears to be insucient to ensure proper convergence to the fundamental eigenstate [START_REF] Fröhlicher | Generational variance reduction in monte carlo criticality simulations as a way of mitigating unwanted correlations[END_REF]. In spite of the unbiasedness of the variance-reduction techniques, the estimates of the critical ux and eective multiplication value in power-iteration calculations are aected by a well-known statistical bias, which is due to weight normalization at the beginning of each generation [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF]. To rst order, the bias is inversely proportional to the number of particles per generation. In Fröhlicher et al.'s case, the number of particles is arguably so small (N = 10 3 ) that the normalization bias cannot be excluded. Moreover, the bias does not necessarily manifest itself in the same way for branchless or branching collisions. Therefore, the bias observed in Ref. 44 cannot be condently attributed to spatial clustering.

In any case, the non-analog methods presented above do have an impact on the generational and spatial correlations of the resulting simulation scheme, and therefore the apparent variance of the sought observables. Our goal is to characterize the eects of collision sampling strategies, variance reduction and population control on correlations, using several kinds of ad hoc estimators, encompassing those that are commonly met for power iterations and some new ones that are hopefully more useful for the interpretation of the simulation results.

. Denition of estimators for correlation diagnostics

We introduce a few relevant estimators for correlation diagnostics. We require all such estimators to be `non-Boltzmann', i.e. they must not be invariant under variance-reduction techniques, otherwise they would not be sensitive to correlations. Their signicance in detecting correlations at the global or local spatial scale will be discussed in the following.

Shannon entropy

The Shannon entropy S of the ssion source is commonly used as a tool for assessing the convergence of power iteration : the power iteration is assumed to have reached the stationary state when S reaches an asymptotic value S ∞ . In view of the fact that the rate of convergence of the Shannon entropy depends on the dominance ratio, and that the dominance ratio also aects the behavior of correlations, the relation between the Shannon entropy of the ssion source and the correlation-induced neutron clustering has been examined by Nowak et al. [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF]. For this purpose, a Cartesian spatial mesh with B cells is superposed over the system geometry and the Shannon entropy S at generation g is computed as customary :

S(g) = - B i=1 p i (g) log 2 (p i (g)),
(5.9)

where p i (g) is the fraction of the total weight of the ssion source particles in the spatial cell i at generation g. In our work, we compute p i (g) as the ratio of the weight of the ssion source particles in cell i over the total weight after normalization. The spatial size of the cells follows from δ = L/B, where L is the linear size of the system. In the simplied homogeneous reactor model tested in Ref. [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF], the occurrence of spatial clustering was shown to lead to lower values of the asymptotic Shannon entropy S ∞ compared to the expected ideal value S id that corresponds to the ideal assumption of N independently distributed particles of unit weight. The extended analysis performed in Ref. [START_REF] Belanger | The eect of branchless collisions on neutron clustering[END_REF] shows that these ndings also apply to continuous-energy transport.

Observe that entropy is just a single scalar for each generation : while it does convey information about the presence of spatial clustering, the ne details of the spatial distribution are lost through the coarse-graining in Eq. (5.9) : only the collective occupation statistics of the cells matters in computing S(g).

Generally speaking, the ideal asymptotic value S id of the entropy function is not known : thus, since the discrepancy between S id and the asymptotic value S ∞ cannot be assessed in practice, the value S(g) resulting from the power iteration cannot be directly used to detect the existence of spatial clustering 2 . Nonetheless, the values of Shannon entropy originating from dierent collision sampling strategies, variance-reduction and population-control methods can be compared relative to each other, which allows assessing the eectiveness of each technique.

Pair distance function

The pair correlation function C(r i , r j , t i , t j ) is a measure of the covariance between the occupation statistics of two spatial sites at positions r i and r j , at time t i and t j , which may coincide. It reads

C(r i , r j , t i , t j ) = E[n i (t i )n j (t j )]
(5.10)

where E[•] denotes the ensemble average, and n i (t i ) is the occupation number in space bin i at time t i , that may be non-integer due to the use of statistical weights. Time can be either continuous (in a time-dependent context) or discrete (i.e. generations). In the context of time-dependent diusionreproduction stochastic processes in innite homogeneous media, Meyer et al. have shown that the pair correlation function of two spatial sites at a given time characterizes the evolution of particle clusters [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. Similar conclusions have been later reached for generalized versions of Meyer's model, encompassing nite-size systems and two observation times [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF][START_REF] Bonnet | Space and time correlations for diusion models with prompt and delayed birth-and-death events[END_REF], and also hold true with minor modications for power iteration, where time t is replaced by discrete generations g [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF][START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF]. While being highly informative, the pair correlation function may be prohibitively expensive to compute for realistic three-dimensional congurations. A convenient alternative consists in collapsing the information content of the pair correlation function into the mean squared pair distance function (often shortened to pair distance function),

⟨r 2 ⟩(g) ∝ |r i -r j | 2 C(r i , r j , g)dr i dr j ,
where C(r i , r j , g) expresses the correlation function for particle pairs at two positions in the same generation, which provides a tool to estimate the typical cluster size [START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. The pair distance function is basically the mean squared distance between any two particles, and can be thus straightforwardly computed in Monte Carlo power iteration, e.g., at the beginning of each generation. Since the number of particle pairs scales quadratically with the number of particles present in the generation, the required computational cost might become an issue ; however, this problem can be mitigated (without incurring any bias) by estimating the mean squared distance from a maximum number of randomly sampled pairs.

Similarly to Shannon entropy, the pair distance function converges towards an asymptotic value ⟨r 2 ⟩ ∞ for a suciently large number of generations, even in the presence of clustering. Departures of ⟨r 2 ⟩ ∞ from the ideal value, corresponding to a simulation with no clustering, would suggest the presence of spatial clustering. If the distribution is uniform and the particles are independent, then it is easy to estimate the ideal value. In general, however, it is impossible to do so. Similarly to the case of the entropy function, the relative values of the mean squared distance can nonetheless be used to compare the eectiveness of dierent variance reduction and population control techniques in quenching the eects of correlations.

The statistics of neutron families

In the initial population at the beginning of the power iteration, one starts with N independent neutrons, each of which is assigned a dierent arbitrary numerical identier. Each identier represents an independent `family' ; the identiers are transmitted to the descendant neutrons at ssion events. Two neutrons sharing the same identier are correlated because they share a common ancestor. During power iteration, the number of neutron families decreases along the generations, because neutrons die and family go extinct by capture or leakage. The intrinsic relationship between diusion-reproduction processes and particle lineages has been explored by Zhang et al. [START_REF] Zhang | Diusion reproduction processes[END_REF] in the context of innite, homogeneous, multiplying media : they showed that all the neutron histories become correlated after a characteristic time proportional to the population size, and they attributed the emergence of neutron clustering to this phenomenon. The eects of neutron lineage statistics on the average size and number of clusters has been recently revisited by Sutton et al. [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF] in the framework of a simplied power iteration model using the WOR sampling method, and in a more realistic multi-group and continuous-energy framework by Belanger et al. [START_REF] Belanger | The eect of branchless collisions on neutron clustering[END_REF] and Frohlicher et al. [START_REF] Fröhlicher | Generational variance reduction in monte carlo criticality simulations as a way of mitigating unwanted correlations[END_REF].

For each independent realization of power iteration, there exists a xation generation at which all the families but one have gone extinct : at the xation generation, all the neutrons are correlated and the population is said to have reached xation. All neutrons now share the same ancestor in the initial source. We derived an exact expression for the average xation time in innite, single-speed homogeneous media in Ref. 84 in the context of kinetic simulations ; these ndings can be straightforwardly extended to processes evolving in discrete generations. We dene η(g) to be the average number of surviving independent neutron families at generation g : the likeliness of occurrence of neutron clustering increases as η becomes much smaller than the initial number of families N .

While the statistics of neutron families does provide interesting insights about correlations, and for this reason it has been proposed as an estimator for clustering [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF][START_REF] Belanger | The eect of branchless collisions on neutron clustering[END_REF], it does not carry information about the spatial and boundary eects (see the discussion in Chapter 4), and this may limit its usefulness, as shown in the following.

Feynman moments

The three previous estimators, although sensitive to correlations, are global and do not provide any information about the local distribution of the correlations within the system. As such, their application to the analysis of correlations occurring in systems exhibiting strong spatial heterogeneity is fraught with conceptual diculties. Ideally, we would like indicators of spatial correlations to be i) local, i.e. spacedependent, and ii) having a known ideal behavior in the absence of correlations, which would make their interpretation easier. For this purpose, we introduce the ensemble Feynman moment

Y O i = V[O i ] E[O i ] , (5.11)
where O i is the value of observable O in spatial cell i. Here E[•] denotes the ensemble average over M independent replicas of a given observable O :

E[O i ] = 1 M M m=1 O i,m , (5.12)
O i,m being the value of the estimator in cell i for replica m. The ensemble unbiased estimator of the variance of O is dened as

V[O i ] = 1 M -1 M m=1 (O i,m -E[O i ]) 2 ,
(5. [START_REF] Kotlyar | Numerically stable Monte Carlo-burnup-thermal hydraulic coupling schemes[END_REF] We stress that V[O i ] is the variance of the estimator and not the variance of the mean of the estimator.

The variance of the mean is obtained by dividing the variance of the estimator by the number of realizations M over which the mean is performed. The Feynman moment has been originally conceived for the detection of ssion-induced correlations in the time series of multiplying systems [START_REF] Pazsit | Neutron Fluctuations: A Treatise on the Physics of Branching Processes[END_REF] : to the best of our knowledge, its application to the analysis of correlations occurring in power iteration has not been attempted before.

As mentioned in Sec. 5.1, to reduce the cost of power iteration, one generally uses ergodic averages over successive neutron generations. We therefore introduce the ergodic estimator of the expectation of O over G active generations, namely,

E[O i ] = 1 G G g=1 O i,g , (5.14)
where O i,g is the value of the estimator in cell i at active generation g. Similarly, one can dene the ergodic estimator of the variance of O by

V[O i ] = 1 G -1 G g=1 (O i,g -E[O i ]) 2 .
(5.15)

Because of positive correlations, we expect V[O i ] < V[O i ] when the two estimators are taken over samples of the same size (G = M ), i.e. when the number of active neutron histories in the single power iteration calculation used to compute the ergodic variance is the same as the sum of active neutron histories simulated throughout all power iteration replicas when computing the ensemble variance [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF].

In order to have a better estimate of Eqs. (5.14) and (5.15), it is possible to perform ensemble averages over M independent replicas on E[•] and V[•]. Finally, we dene the ergodic Feynman moment of O by

Y G O i = E[V[O i ]] E[E[O i ]]
, (5.16) which reduces to the ensemble Feynman moment dened in Eq. (5.11) when G = 1.

Since the Feynman moment for Poisson variates is equal to 1, it is possible to interpret its value as the deviation of a given observable (in a spatial cell) from a Poisson-like behavior. A process that generates Poisson-distributed counts is the following : consider N particles, independently and identically distributed into K bins following the probability distribution p. If p k is the probability that a particle falls within bin k, then if ∀k, p k ≪ 1, the particle counts are approximately Poisson distributed in each cell, and the Feynman moment Y k for cell k is approximately equal to 1, for all the bins and independently of the probability distribution p. However, when there are very few populated bins, then p k ∼ 1 for some k (i.e. the probability that a particle falls within bin k is large) ; in this situation, the counts actually follow a binomial distribution, and we expect Y < 1. Finally, suppose that the number of bins is high enough so that the number of particles in each bin is Poisson distributed. The particles are divided into groups of size α, and each group is independently tossed into a bin. In this situation, it is easy to check that the Feynman moment is equal to α, again for all the bins and independently of the probability distribution. This suggests that the value of the Feynman moment for a counting variable can be interpreted as the typical `weight' of a particle cluster. This example also suggests that Feynman moments are most naturally applied to quantities that are akin to counts. Indeed, for counting variables, the Feynman moment is dimensionless and has a natural reference value to which it can be compared.

It is less naturally applicable to commonly used quantities such as uxes, which is why such quantities are not considered in the present work.

In this work, we apply the Feynman moments to the collision counts ψ, i.e. the average number of neutrons entering a collision in a space bin 3 , and to the ssion emission counts F , i.e. the average number of ssion neutrons being emitted in a space bin. We will focus exclusively on energy-integrated Feynman moments.

Normalized variance

Another local estimator for correlations can be introduced using the ensemble estimator for the normalized variance

g O i = V[O i ] E[O i ] 2 ,
(5.17

)
where E[O i ] and V[O i ] are dened as above. The normalized variance describes the amplitude of uctuations relative to the average and is a simplied form of the centered normalized pair correlation function that is widely used in the context of spatial clustering analysis [START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF][START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF][START_REF] Meyer | Clustering of independently diusing individuals by birth and death processes[END_REF]. Observe that the normalized variance diers from the Feynman moment only in the normalization terms appearing at the denominator.

However, this dierence is crucial and conveys a distinct information content with respect to correlations, as illustrated in Sec. 5.4. The ergodic normalized variance g G O i can be dened similarly to that of the Feynman moments, namely

g G O i = E[V[O i ]] E[E[O i ]] 2 .
(5.18)

In this work, we apply the normalized variance to the same observables as for the Feynman moments, i.e. the energy-integrated collision counts and ssion emission counts.

. Analysis of a simple benchmark with single-speed particle transport

A previous study by Nowak et al. looked at the eects of particle clustering on the Shannon entropy of the ssion source [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF]. For a simple benchmark model of single-speed neutron transport in a uniform cubic reactor, with reective boundary conditions, the fundamental eigenmode of the ssion source is spatially uniform. In an analog simulation without clustering, the positions of the N ssion neutrons at equilibrium would all be independent and uniformly distributed. Thus, we can decompose the cube into B elements of equal volume and measure the degree of clustering by comparing the entropy of the ssion

S id = log 2 (N ) - B N B -1 B N N k=1 (B -1) -k N k k log 2 (k).
( To evaluate the possible benets of using branchless collisions over the more traditional branching algorithm, we have replicated the study performed by Nowak et al., performing k-eigenvalue power iteration for single-speed neutron transport in a reected, critical cubical reactor, with side lengths of 400 cm, and cross sections of Σ s = 0.27 cm -1 , Σ c = 0.02 cm -1 , ν f = 2.5, and Σ f = Σc ν f -1 . The total weight of the particles at the beginning of each generation is N = 10 3 , and the initial source distribution is a point located at the center of the cube, with source particles having unit weight. The aim of this simulation is to provide a rst indication of whether the branchless collision method can reduce the clustering of the particles in the simulation, compared to traditional branching transport. The simulations of three-dimensional congurations presented in this section were performed using Chenille, a mini-app also developed at CEA [START_REF] Belanger | Development of numerical methods for neutronics in continuous media[END_REF]. In Chenille, the formula used to determine the number of ssion progeny when using branching collision is

n f = w k (g-1) e • ν(E)Σ f (E) Σ t (E) + ξ , (5.20)
where k g-1 e denotes the estimate of the multiplication factor stemming from the previous generation ; in this scheme, ssion particles are born with unit weight. For the remainder of the present section and for Sec. 5.5, we shall use this formula instead of Eq. (5.3).

The entropy of the ssion source for single-speed transport in the critical cube is plotted in Fig. 5.1, as a function of the ssion generations, for both branching and branchless simulations, along with the theoretical value for the ideal case of perfectly independent and homogeneously distributed particles.

Clustering in the branching simulation leads to a ssion source entropy which is much lower than the value associated with independent neutron histories, for the given number of particles, and entropy mesh. Using branchless collisions, however, the entropy of the ssion source converges to the ideal value, and experiences much smaller uctuations. This benchmark conguration is unique for several reasons. compared to the traditional branching algorithm. In contrast with the behaviour observed for the critical system, the source entropy of the branchless collision does not reach the ideal entropy (cf. Fig. 5.1).

The reason for this is that, once the system is no longer exactly critical, population control mechanisms are required, which always causes the number of particle families in the simulation to decrease to 1 (xation). The number of generations to xation for the branching algorithm is much smaller than for branchless algorithms. Furthermore, for both the subcritical and supercritical cases, branchless collisions with splitting for population control and a low roulette threshold seem to give the longest xation time.

With branching transport, the xation time is generally comparable to the time required for the ssion source to converge, while the xation time for the branchless algorithms is much longer than the time required for entropy convergence. Surprisingly, sampling WOR on its own had very poor performance when used with branchless collisions, in apparent contradiction with Sutton [42]. This discrepancy is likely attributed to the fact that, in Sutton's algorithm, each absorption site may contribute to populate the ssion bank, which potentially induces less variation in the particle weights. Sampling WOR in conjunction with splitting had reasonable performance, but was still not as ecient as using splitting alone.

. Specications for more complex benchmark problems

In order to gain a deeper insight, we have extended the simple system investigated in the previous section by introducing a set of benchmark problems, of which we provide the key specications.

. Description of the chosen congurations

For the purpose of our analysis, we have chosen a total of four one-dimensional critical slab geometries taken from Vitali et al. [START_REF] Vitali | Eigenvalue separation and eigenmode analysis by matrix-lling Monte Carlo methods[END_REF]. The selected congurations have dierent values of dominance ratio R, and cover both homogeneous and heterogeneous systems, which are known to behave dierently with respect to the parameter R. for the moderator. The nuclear data for all the materials are intended to be realistic, and are represented using a three-group formalism, with thermal, epi-thermal and fast energy groups. In Ref. 1, prompt χ p and delayed χ j d spectra for each precursor family j were provided separately. Here we collapsed them into an average ssion spectrum χ, with χ = (ν p χ p + j ν j d χ j d )/ν.

Following Vitali et al., for all congurations we modify the capture cross section of the homogeneous material (for cases 1 and 2) and of UO 2 (for cases 3 and 4) so that the systems are critical. The system- dependent capture cross-sections are given in Table 5.5. The dominance ratios for each conguration have been computed by Vitali et al. and are also recalled in Tab. 5.1.

Parameters

g = 1 (fast) g = 2 (epithermal) g = 3 (thermal) Σ f,g [cm -1 ] 3.0586 × 10 -3 2.1579 × 10 -3 5.6928 × 10 -2 Σ s,1→g [cm -1 ] 4.48187 × 10 -1 1.78483 × 10 -1 0 Σ s,2→g [cm -1 ] 0 1.8058111 1.41764 × 10 -1 Σ s,3→g [cm -1 ] 0 0 4.31567 ν g [-]
2.4 2.4 2.4 χ g [-] 0.876304 0.123696 0 Table 5.2 : Nuclear data of the homogeneous fuel-moderator material. The three energy groups are indexed by g.

Parameters g = 1 (fast) g = 2 (epithermal) g = 3 (thermal) Σ f,g [cm -1 ] 3.0586 × 10 -3 2.1579 × 10 -3 5.6928 × 10 -2 Σ s,1→g [cm -1 ] 2.21062 × 10 -1 7.3843 × 10 -2 0 Σ s,2→g [cm -1 ] 0 7.77642 × 10 -1 4.3803 × 10 -2 Σ s,3→g [cm -1 ] 0 0 1.55272 ν g [-] 2.4 2.4 2.4 χ g [-]
0.876304 0.123696 0 Table 5.3 : Nuclear data of the UO 2 material. The three energy groups are indexed by g.

. Description of the sampling strategies

Five algorithms (i.e. combinations of collision sampling strategies and population control algorithms) will be considered for each geometry : i) branching collisions ; ii) last-event estimator with sampling WOR ; iii) branchless collisions ; iv) branchless collisions with weight combing ; v) branchless collisions with sampling WOR. Russian roulette and splitting are always applied to the particles emerging from collision events, with roulette threshold w R = 0.8 and unit target weight, and splitting threshold w S = 2.

A neutron of statistical weight w is split into n S = ⌊w/w S ⌋ particles with equal statistical weight w/n S .

The choice of these (somewhat arbitrary) thresholds is inspired by those used in the TRIPOLI-4

® code developed at CEA [START_REF] Brun | TRIPOLI-4®, CEA, EDF and AREVA reference Monte Carlo code[END_REF]. It should be noted in particular that the choice of w S can have a signicant eect on the neutron correlations, especially if w S is too small. Finally, when no population control algorithm is specied, the ssion bank is wholly transmitted to the next generation ; in particular, no sampling with replacement is applied.

. Simulation results

In the following, we present numerical results for the benchmark congurations described in Sec. 5.3

and the ve algorithms presented in Sec. 5.1, rst for the global tallies (entropy, pair distance and family number), and then for the local tallies (Feynman moment and normalized variance). The aim of this investigation is to assess the impact of each algorithm on correlations, with the help of the diagnostics tools introduced in Sec. 5.1.3. All the power iteration simulations carried out here start with an initial neutron source belonging to the fast group and uniformly distributed in space (non-ssile media included).

The number of inactive generations for each conguration is given in Tab. 5.1 and has been determined on the basis of Fig. 5.4, while the number of neutrons per generation will be detailed for each case separately.

Parameters

g = 1 (fast) g = 2 (epithermal) g = 3 (thermal) Σ c,g [cm -1 ] 3.05 × 10 -4 3.699 × 10 -4 1.825 × 10 -2 Σ s,1→g [cm -1 ] 2.27125 × 10 -1 1.0464 × 10 -1 0 Σ s,2→g [cm -1 ] 0 1.02817 9.7961 × 10 -2 Σ s,3→g [cm -1 ] 0 0 2.76295
Table 5.4 : Nuclear data for the H 2 O material. The three energy groups are indexed by g. Table 5.5 : Capture sections Σ c,g for each configuration. The three energy groups are indexed by g.

System Number

Σ c,1 [cm -1 ] Σ c,2 [cm -1 ] Σ c,3 [cm -1 ]

. Analysis of global estimators

We begin our analysis by considering global tallies. The evolution of the Shannon entropy S(g) as a function of the generations in each of the benchmark congurations is illustrated in Fig. 5.4. As expected, all entropy curves converge towards an asymptotic value for a large number of generations.

The asymptotic value of the entropy depends on the algorithm and the system, whereas the convergence rate (i.e. the number of generations needed to attain convergence) only depends on the system. For systems 1, 2, and 4, the asymptotic value of the entropy is smaller for the branching algorithm than for any of the other algorithms, suggesting stronger spatial correlations when using branching collisions.

The method relying on the last-event estimator taken from Ref. 42 leads to an entropy lying between that of the branchless algorithms with and without population control. The typical spatial patchiness induced by spatial correlations is illustrated in Fig. 1.1 for the case of system 2 (homogeneous reactor with large dominance ratio). It is interesting to note that in system 3 all algorithms lead to similar asymptotic entropy ; additionally, the branching algorithm appears to lead to the higher entropy, and is thus expected to induce less spatial correlations. The behavior of system 3 can be probably explained by its small dominance ratio, cf. Tab. 5.1. Spatial clustering is the result of a competition between diusion and multiplication : correlated branches induce spatial correlations when the neutrons cannot explore the whole system. When using branchless collisions, the only way for correlated branches to appear is through splitting or population control. In systems close to criticality such as ours, the weight correction factor in branchless algorithms is close to unity, which means that splitting seldom occurs during a generation.

However, as stated previously, the number of neutrons decreases from one ssion bank to the next. Thus, weight normalization increases the average weight of neutrons, ultimately leading to additional splitting.

Alternatively, if population control such as combing or sampling WOR is applied, then the same neutron in the ssion bank may be copied several times, introducing additional correlated branches.

Careful inspection of Fig. 5.4 will reveal that the dierence between the asymptotic entropy of the branching algorithm, and of the four other algorithms increases with the dominance ratio. Conversely, if the dominance ratio is small enough, the entropy for branching collisions becomes slightly higher than for branchless collisions, as illustrated by Fig. 5.4c for system 3. In this case, correlated branches explore the whole viable phase space, and the eects of spatial clustering will be quenched. Said dierently, neutrons produce more ssion neutrons (i.e. correlated branches) when using branching collisions than branchless collisions, meaning that more random walkers are available to explore the phase space. Given that spatial correlations are weak, the increasing number of events leads to better estimates of scores. This behavior is not specic to heterogeneous systems (such as conguration 3) : we have found similar results for a homogeneous conguration with a dominance ratio close to that of system 3. Note however that, in systems with small dominance ratios, the dierence among the asymptotic entropy values for the ve algorithms is rather small, translating the fact that systems with small dominance ratio are anyway only weakly aected by spatial correlations and, thus, by clustering.

We conclude our analysis of Shannon entropy with a discussion on the dependence of the asymptotic entropy (i.e. the entropy reached after the inactive generations) on the spatial cell size, shown in Figs. 5.5.

The dependence of S ∞ on the bin size is similar in both homogeneous and heterogeneous systems. As suggested by Fig. 5.4c, the Shannon entropy for the branching algorithm exceeds that of the other algorithms when the cell size is very small, while it is lower or equal for larger cell sizes. In addition, these trends appear to be strengthened in systems with large dominance ratio, suggesting that the eect of spatial correlations becomes irrelevant when the bin size becomes small enough. In this case, only the number of available random walkers matters, and using branching collisions might become advantageous.

In any cases, the impact of the bin size on the value of the entropy is much stronger than the impact of the algorithm.

The indications that can be drawn from Fig. 5.6 for the pair distance function are consistent with those stemming from the entropy tally, with one signicant exception : contrary to entropy, population control seems to have a weak inuence on the pair correlation function. Moreover, Fig. 5.6c shows that the pair distance for the branching algorithm in system 3 is the lowest amongst the ve algorithms considered here (the dierences are nonetheless small). This is in contrast with the intuition that lower entropy values are the signature of enhanced spatial clustering. We showed above that, for system 3, the highest entropy (and thus, the situation with the smallest impact of neutron clustering) was also achieved for the branching algorithm. Consequently, we would have expected that the largest pair distance function would be attained for the branching algorithm. Nonetheless, system 3 will not exhibit signicant clustering for any reasonable number of particles, mostly because of its low dominance ratio.

For systems exhibiting strong heterogeneity, such as system 4, the use of the average pair distance tally to assess the impact of spatial correlations in the ssion source becomes questionable, since the moderator region does not contain ssion neutrons : if the pair distance is comparable to the linear size of the moderator region, then it is unclear whether the pair distance simply encodes the distance between the two ssile regions. although the associated entropy is the highest amongst the algorithms considered here. This surprising nding is coherent with previous work : the statistics of families are unsuitable to detect correlations in systems of nite size [START_REF] Bonnet | The statistics of family histories in kinetic Monte-Carlo simulations[END_REF]. In innite media, the dynamics of spatial correlations is driven by the statistics of families, due to the lack of a spatial scale. On the contrary, in systems of nite size, the emergence of spatial clustering depends on the competition between the number of generations required to reach xation and the number of generations required to reach mixing, i.e. to explore the whole space [START_REF] De Mulatier | The critical catastrophe revisited[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF][START_REF] Bonnet | Space and time correlations for diusion models with prompt and delayed birth-and-death events[END_REF]. In this context, the statistics of families alone becomes insucient to provide the full picture.

. Analysis of the ssion source : Feynman moment

In view of the shortcomings of global tallies, we move now to the analysis of local tallies : the Feynman moment and the normalized variance. These tallies will be applied to the ssion source and to the collision counts. We will investigate both the spatial shape of the local tallies and their dependence on the mesh cell size δ. Indeed, coarse-graining over a length δ inuences the impact of correlations on the sought observables.

The Feynman moment of the ssion source describes the correlations between ssion neutrons.

Figure 5.8 shows the ensemble Feynman moments for the ssion source after the inactive generations, for each conguration and algorithm considered. Let us initially focus on the homogeneous cases, i.e. systems 1 and 2. In the ideal case where all particles are independent, we expect ssion neutrons to be Poisson distributed, which is equivalent to a spatially at Feynman moment Y ≃ 1 everywhere.

Figures 5.8a and 5.8b show that the Feynman moment for the branching algorithm in a homogeneous conguration has a non-trivial shape, indicating that deviations from the ideal behavior do not develop uniformly. The same gures illustrate that for this particular choice of bin size δ, simulations using branchless collisions without additional population control introduce smaller and more evenly distributed uctuations in the distribution of ssion sites, as witnessed by the lower and spatially atter Feynman moment. The discrepancy in the Feynman moment for branchless and branching collisions increases with increasing dominance ratio : for branching collisions Y increases with increasing dominance ratio, whereas it is fairly constant for branchless collisions when increasing the dominance ratio. Introducing combing or sampling WOR in branchless algorithms further lowers the Feynman moments and introduces a slight spatial shape, which suggests that population control algorithms amplify the eects of leakage boundary conditions on spatial correlations. This is reminiscent of what happens with reecting boundary conditions [START_REF] Sutton | Application of a discretized phase-space approach to the analysis of Monte Carlo uncertainties[END_REF]. The last-event algorithm shares the same spatial shape as the branchless collisions with population control algorithms, with a slightly larger value, suggesting that it is slightly less eective in quenching spatial correlations.

Insight can be gained by examining how Y scales with respect to the cell size δ. We dene l clust as the typical size of the largest cluster in the system. In systems that are homogeneous or where the eects of heterogeneity on particle transport are mild, the typical cluster size can be estimated using l clust ∼ ⟨r 2 ⟩ ∞ ; this assumption fails for systems where the eects of heterogeneity are more pronounced, such as conguration 4. We thus have the following picture : if δ ≃ l clust , the observables will be strongly aected by correlations ; on the contrary, if δ ≪ l clust , the probability that two particles belonging to the same cluster fall into the same cell becomes small, and the eects of the correlations will be milder.

Y av F = 1 B ′ B ′ i=1 Y F i , (5.21)
where B ′ is the number of spatial cells with ssile material. As we consider homogeneous systems, l clust ∼ ⟨r 2 ⟩ ∞ . The Feynman moment for branching collisions has a power law increase for δ ≲ ⟨r 2 ⟩ ∞ and saturates when δ ∼ ⟨r 2 ⟩ ∞ . This translates the fact that there exists clusters at all spatial scales up to ⟨r 2 ⟩ ∞ . More strikingly, the Feynman moment for branchless collisions without population control (a) Y F in power iteration for system 1 with 10 5 independent replicas.

(b) Y F in power iteration for system 2 with 10 5 independent replicas.

(c) Y F in power iteration for system 3 with 10 4 independent replicas.

(d) Y F in power iteration for system 4 with 10 4 independent replicas. is constant when δ ≲ ⟨r 2 ⟩ ∞ and decreases when δ ∼ ⟨r 2 ⟩ ∞ , the latter being related to the counts following a binomial law instead of a Poisson law when the number of bins is small, as mentioned earlier.

In this case, clustering is completely mitigated by the use of branchless collisions : if ssion neutrons are Poisson distributed, the variance and the average of the distribution of ssion neutrons scale similarly with respect to δ and the Feynman moment becomes constant. When adding combing or sampling WOR, the Feynman moment stays constant for δ ≪ ⟨r 2 ⟩ ∞ and start increasing when approaching δ ∼ ⟨r 2 ⟩ ∞ , whereupon it saturates. We observe the same behaviour for the last-event algorithm, with larger values.

The shape of the curves in Fig. 5.9 can be given an interesting interpretation. When the average Feynman moment is constant for small values of δ, as for all algorithms except the branching algorithm, it translates the fact that clusters have some minimum spatial scale ; at scales ner than the minimum cluster size, clusters are essentially innitely large, and the neutrons occupation statistics behaves as if the particles were independent. For very ne meshes, branching collisions appear to be more ecient than branchless collisions in quenching correlations. This could be explained by the fact that, as the cell size decreases, the probability that correlated particles fall within the same cell also decreases ; the fact that neutrons are strongly correlated when using branching collisions becomes irrelevant, but the fact that there are many more particles (thus, more sampling events) than when using branchless collisions or the last-event algorithm holds true. At the opposite extreme, the Feynman moment saturates at scales of the order of δ ∼ ⟨r 2 ⟩ ∞ , which (at least in homogeneous systems) can be interpreted as the typical size of the largest clusters. Between the minimum and maximum scales, the Feynman moment follows a power law, which is the expected behavior for self-organized critical systems, as recently suggested by Dechenaux et al. on the basis of the application of techniques borrowed from the renormalization group in particle physics [START_REF] Dechenaux | Percolation properties of the neutron population in nuclear reactors[END_REF]. The power-law exponents range between 0.5 and 1 for systems 1 and 2. The occurrence of power laws in the scale dependence of Feynman spectra is suggestive, but more careful analysis is required to conrm the existence of a deeper connection. Finally, at scales coarser than the typical cluster size, the occupation statistics of the space bins behaves as if the system were populated by independent macro-particles. In the limit δ → L, the Feynman moment decreases, as the counts follow binomial laws, and converges to zero because of the constraint on weight conservation of the ssion source.

We move now to the case of heterogeneous systems 3 and 4. Figure 5.8c shows the Feynman moments for system 3, which has a small dominance ratio. For this case, all algorithms behave similarly.

The main dierence lies at the fuel-moderator interfaces. Here, branchless collisions lead to a dip in the Feynman moment near the material boundary, and a peak for branching collisions, while the last-event algorithm leads to an almost at spatial shape. Figure 5.8d shows that for case 4, which has a larger dominance ratio, the Feynman moment is signicantly smaller when using branchless collisions than when using branching collisions. In both systems the use of the last-event algorithm leads to a larger value of Feynman moments than when using any of the branchless variants. In system 4, this value is nonetheless much smaller than when using branching collisions. The spatial shape for all Feynman moments is similar in system 4, suggesting that spatial correlations are always at play, although they are the weakest for branchless collisions.

Again, interesting conclusions can be drawn from the dependence of the Feynman moment on δ. First, due to heterogeneity, l clust cannot be identied with the pair distance. However, we still use the pair distance as a qualitative estimate of l clust . Figure 5.9c shows that the spatially-averaged Feynman moment for all branchless variants in the heterogeneous conguration with a small dominance ratio is roughly constant for δ ≲ ⟨r 2 ⟩ ∞ , and decreases for δ ≳ ⟨r 2 ⟩ ∞ . This suggests the absence of strong spatial correlations when using any of the ve algorithms ; it also illustrates that the pair distance function is still a good approximation of the typical cluster size for systems with mild heterogeneity. On the other hand, Fig. 5.9d shows that Y av F ∝ δ 0.82 for δ ≲ ⟨r 2 ⟩ ∞ for all algorithms in the heterogeneous conguration with a large dominance ratio. Moreover, the power law scaling of the space-averaged Feynman moment as a function of δ is the same for each algorithm. By analogy with the three other cases, we expect the space-averaged Feynman moment in system 4 to reach a maximum for δ ≳ l clust ; a direct conrmation is not possible, because ⟨r 2 ⟩ ∞ is essentially driven by the size of the moderator region, which means that it is not a good approximation of the (unknown) typical cluster size.

So far, we have investigated ensemble Feynman moments, which are not directly accessible by power iteration (unless one accepts to sample independent replicas). One may wonder how the ergodic Feynman moment relates to the ensemble Feynman moment. We consider then the dependence of the Feynman moment on the number of active generations G : based on Refs. 36 and 45, we expect that, when the number of active generations is large enough, the apparent variance converges towards the real variance.

Therefore, we also expect the ergodic Feynman moments to converge towards the ensemble Feynman moments. In what follows, we want to characterize the relationship between the value of the dominance ratio and the ability of our algorithms to mitigate the bias on the variance induced by the use of ergodic averages instead of ensemble averages. We stress this bias is not related to discarding an insucient number of inactive generations before tallying the observables. ratio. However, the `initial oset' depends on sampling and population control. This is reminiscent of the fact that, in power iteration, the speed of convergence of the dominant eigenvalue of k 0 only depends on the dominance ratio, but the initial value of k 0 (and so the number of inactive generations to discard) depends on the initial condition. In addition, the eect of population control algorithms appears to be milder compared to the eect of collision biasing.

We conclude our analysis by considering the eects of the neutron population size. Previous investigations have shown that the impact of spatial clustering becomes milder as the number of particles per generation increases [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. show the space-averaged ensemble Feynman moments for all ve algorithms in system 2 (homogeneous case with higher dominance ratio), as a function of the population size. We observe that the Feynman moment for the branching algorithm converges towards an asymptotic value for large N . This scaling is consistent with similar ndings in the context of time-dependent Monte Carlo simulations, for the case of spatially homogeneous branching-diusion processes under population control [START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF]. Additionally, we veried that the shape of the Feynman moment also converges to an asymptotic shape for large N (not showed here for conciseness). We further observe that, using branchless collisions as well as using the last-event algorithm in systems 1-3, Feynman moments appear to be almost independent of N (only shown for system 2 in Fig. 5.11a). This behavior is presumably due to the absence of signicant spatial correlations with these algorithms. Conversely, in system 4, where strong spatial correlations exist for all algorithms, the Feynman moments depend on N and converge towards an asymptotic value for large N . It should be noted that, even then, the Feynman moment of branchless variants and of the last-event algorithm saturates at a smaller value of N than for the branching algorithm.

. Analysis of the ssion source : normalized variance

Additional insight can be obtained by assessing the behavior of the normalized variance g F of the ssion source. As discussed above, g ∼ 0 is a reasonable denition of `weak correlations', similarly as for Y ∼ 1. However, in the following we will show that the two tallies are not equivalent, and convey dierent pieces of information. The amplitude of ssion-induced spatial uctuations depends on the population size, and we will illustrate that this is captured by normalized variances.

All cases in Fig. 5.12 display a signicant increase of g F close to the leakage boundary conditions, even when using branchless collisions. This peak is explained by inspecting Fig. 5.8 : although Feynman moments are (almost) at for branchless collisions, the average of the ssion source is close to zero near leakage boundary conditions. Since g F = Y F /E[F ], this explains the shape of g F .

Let us analyze the homogeneous systems rst. In Figs. 5.12a-5.12b we observe that the normalized variances of all algorithms except the branching algorithm are rather similar, and close to zero in the bulk of the geometry. On the other hand, the normalized variance using the branching algorithm can be close to 1 even far from the boundaries, which suggests that large uctuations will occur. We observe that the shape of the normalized variance for all algorithms is almost independent of the dominance ratio. In the case of the heterogeneous media, Fig. 5.12c for case 3 shows that all the algorithms behave similarly (a) For homogeneous configuration with large dominance ratio (system 2) (b) For heterogeneous configuration with large dominance ratio (system 4) in terms of normalized variances. Furthermore, Fig. 5.12d for case 4 conrms the hierarchy between the algorithms, and shows that, even in the bulk of the system, typical uctuations are larger for the branching algorithm than for the other algorithms. Similarly to the homogeneous cases, the normalized variance is only weakly dependent on the dominance ratio when using branchless algorithms or the last-event algorithm.

Next, we examine the dependence of normalized variances on the population size N and the cell size δ. For the sake of conciseness, we will not consider the dependence on G, the number of active generations, because these eects are very similar to those occurring for Feynman moments of Sec. 5.4.2. As for the dependence on δ, previous ndings suggest that g F ∼ 1/δ for small values of δ [START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF]. Figure 5.13 illustrates this scaling. We consider

g av F (B) = 1 B ′ B ′ i=1 g F i , (5.22)
where B ′ is the number of cells with ssile material. We note that g av F ∝ B for all algorithms, when L/δ = B ≫ 1. For large cell sizes, the normalized variance is smaller for branchless collisions than for branching collisions. The converse is true for small cell sizes. This can be explained by the fact that spatial correlations are weakened by decreasing the cell size. Branching sampling typically results in more ssion neutrons than branchless sampling, because each collision site is a potential ssion site in branching sampling. Due to the excess of ssion neutrons, when spatial correlations are weak, the branching algorithm introduces less variance in the estimation of the ssion source than branchless algorithms. There are essentially two ways to obtain weak spatial correlations : either the dominance ratio is small enough (cf. system 3) or the cell size is small enough (as clearly illustrated by Figs. 5.13b and 5.13d). Finally, additional numerical ndings (not shown here for the sake of conciseness) suggest that the normalized variances scale as 1/N regardless of the algorithm, which is consistent with the literature [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

. Analysis of the collision counts : Feynman moment

Most of the tallies of interest for reactor physics (such as reaction rates or energy deposition) are estimated by summing over all the collisions in a generation. In this respect, the Feynman moments of the collision counts can help to interpret the behavior of spatial correlations for such observables.

The Feynman moments of the collision counts, shown in Fig. 5.14, are signicantly higher than those of the ssion source. Scattering now contributes to the tally and there are several scattering events per generation ; in contrast, there is only one ssion event per generation. Additionally, correlations between collision sites are stronger than correlations between ssion sites. Figures 5.14c the heterogeneous cases 3 and 4 indicate strong spatial correlations between collision sites near the fuelmoderator interfaces. Similar peaks in the spatial prole of the correlations were already observed in the ssion source, but their amplitude is considerably amplied in the Feynman moments of the collision counts. Far from the ssile regions, all algorithms behave similarly, due to the fact that they all reduce to implicit capture in the moderator region. Note that sampling WOR has no signicant eect on the Feynman moments of the collision counts. Figures 5.15a and 5.15b show the dependence of the spatially-averaged Feynman moments of the collision counts with respect to the bin size δ, in homogeneous and heterogeneous systems, respectively. All the Feynman moments of the collision counts scale as a power law of δ, for δ ≲ l clust . Additionally, Fig. 5.15a saturates around ⟨r 2 ⟩ ∞ , similarly to the Feynman moment of the ssion source. The latter remark does not hold for system 4 (see Fig. 5.15b), which is related to the discussion on Fig. 5.9d (in Section 5.4.2), where we observed the same trend for the dependency on δ of the Feynman moments for the ssion source. In any case, branchless collisions lead to Feynman moments that are considerably smaller than those of branching collisions, and all population control algorithms seem to have little eect on the dependence on coarse-graining of the Feynman moments of the collision counts. The dependence on N and G of the Feynman moment of the collision counts is similar to that of the ssion source, and for the sake of conciseness it will not be shown here.

. Analysis of the collision counts : normalized variance

In the homogeneous cases, the normalized variance of the collision counts behaves similarly to that of the ssion source (not shown here for conciseness). The two heterogeneous congurations are shown in Fig. 5. [START_REF] Tuominen | Eect of energy deposition modelling in coupled steady state Monte Carlo neutronics/thermal hydraulics calculations[END_REF]. In both cases 3 and 4, it is interesting to note that the peaks observed in the corresponding Feynman moments do not occur in the normalized variance. This illustrates clearly the dierence between Feynman moments and normalized variances.

Finally, we consider the dependence of normalized variance with the size of spatial cells, which is expected to scale as 1/δ when δ is small enough [START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF] ; indeed, Fig. 5.17 shows that the normalized variance behaves as 1/δ. When the dominance ratio is close to unity (see Fig. 5.17d), the normalized variance for the branching algorithm is signicantly higher than for the other algorithms. Additionally, although all algorithms except from the branching algorithm yield similar normalized variances, branchless collisions in combination with either combing or sampling WOR appear to consistently yield the lowest normalized variance in our systems.

. A digression on continuous-energy problems

Most investigations concerned with possible methods to reduce clustering typically consider simplied systems in a single-speed or multi-group framework [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF], such as those examined in Sec. 5.2 and 5.4. It is indeed easier to draw conclusions on the performance of these systems and methods when disregarding heterogeneity and continuous energy ; however, in realistic applications, the eects of these two aspects cannot be neglected. In Sec. 5.4, we discussed the eect of heterogeneity on spatial correlations. Now, we shall investigate how continuous energy aects the statistical behaviour of the neutron population. Interaction probabilities now depend on the isotopic concentration in materials, so one cannot indiscriminately use microscopic or macroscopic cross sections. In the original formulation of the branchless collision, however, it is somewhat ambiguous whether the cross sections are intended to be the macroscopic cross sections for the material, or the microscopic cross sections for the sampled isotope [START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF]. There are in fact several ways to sample branchless collisions. One can sample rst the collision channel (ssion, capture or scattering), whereupon the weight multiplier is independent of the chosen isotope, or one can rst sample the collision isotope, whereupon the weight multiplier is independent of the collision channel.

Additionally, one can also choose whether to use the cross section or the yield-weighted cross section for making the rst sampling (be it for the isotope or channel). Here, we will briey outline two options for this sampling procedure. Note that the distinction between these two methods only makes sense when considering continuous energy. In a single-speed (or multi-group) system, where there is eectively just one isotope per material, there is no dierence between performing branchless collisions on the isotope, or on the material.

. Branchless collisions on the isotope

We begin by illustrating the branchless collision strategy applied to an isotope. In most continuousenergy Monte Carlo codes, the collision event begins with the sampling of isotope i, whose probability is given by

P i (r, E) = N i (r) σ i,t (E) Σ t (r, E) , (5.23)
where N i (r) is the concentration of isotope i at position r, σ i,t (E) is the total microscopic cross section for isotope i at incident energy E (see Chapter 2 for details). Due to this simple fact, developers interested in adding branchless collisions to a pre-existing Monte Carlo code are likely to immediately consider this algorithm, which avoids a potentially large refactoring of source code. It is possible to keep the methods for sampling the isotope, and only change how the reaction channel is sampled. This is why the isotopebased branchless collision method was implemented in TRIPOLI-4 ® , when the reactor kinetics simulation was added to the code [START_REF] Faucher | New kinetic simulation capabilities for Tripoli-4®: Methods and applications[END_REF].

Isotope i is rst sampled using Eq. (5.23). Knowing the isotope i, the scattering and ssion proba- bilities follow from

P s|i = σ i,s (E) ν i,f (E)σ i,f (E) + σ i,s (E) and P f |i = ν i,f (E)σ i,f (E) ν i,f (E)σ i,f (E) + σ i,s (E) . (5.

24)

If scattering is chosen, the scattering channel α is then sampled in the standard manner, with probability

P sα|i = σ i,sα (E) σ i,s (E)
.

(5.25)

After the nal reaction state has been determined, the particle weight is modied by

w ′ = w ν i,f (E)σ i,f (E) + σ i,s (E) σ i,t (E)
.

(5.26)

Note that this form implicitly assumes that the yield of all scattering reactions is exactly 1. Explicitly considering the yields of reactions such as (n, xn), which are somewhat rare, would add to the computational complexity when computing the scattering and ssion probabilities, and change how scattering reaction channels are sampled. Instead, here we assume that the scattering yield is always 1 for the probabilities and weight modier in Eqs. (5.24) and (5.26) ; if we sample a scattering reaction where ν i,sα (E) ̸ = 1, we also multiply the neutron weight by ν i,sα (E). It should also be noted that while our weight modier in Eq. (5.26) is independent of the reaction channel, it is dependent on the sampled isotope at the collisions site. Therefore, at the same position in phase space, this weight multiplier could exhibit a large variance ; in the fuel for example, all the isotopes but one could have a null ssion cross As the weight modier for the single-speed critical conguration is always equal to one, all the particles are able to remain truly independent, eliminating the possibility of clustering. While the continuousenergy case is nearly critical, the weight multiplier is never equal to one, and depends on the energy of the incident particle (for both the isotopic and material variants). This causes the weights of the particles to diverge from unity, and the use of population control mechanisms is then required. Combing and splitting both lead to the eective duplication of particles at a phase-space coordinate, and therefore contribute to clustering. Branchless collisions on the isotope can in fact exacerbate this situation, as the weight multiplier could vary greatly between dierent isotopes in a material. Branchless collisions on the material reduce the variance of the weight multiplier, because the latter is independent of the isotope and the reaction channel. Thus, the number of particle families decreases more slowly ; particle weights are less dispersed, and therefore less population control is needed.

. Conclusions

In this chapter we have investigated the impact of sampling strategies, variance-reduction and population-control methods on the correlations that occur in the Monte Carlo implementation of the power iteration algorithm, which is key to the sampling of the critical source for kinetic simulations. For this purpose, we have selected several distinct tallies : the Shannon entropy, the average pair distance squared, the average number of surviving families, the Feynman moments and the normalized variance. This analysis has been carried out on a set of simple, yet meaningful benchmark congurations encompassing homogeneous and heterogeneous geometries. In order to probe the eects of each technique, the power iteration was run with dierent combinations of branching and branchless collisions as well as the last-event estimator for the ssion source, and combing or sampling WOR.

All the proposed tallies help in some way to detect the presence of anomalous uctuations in the examined systems. However, each comes with specic advantages and drawbacks. Global (i.e., integrated) tallies such as the entropy and the average square pair distance, while being easy to use and interpret, have been shown to be inadequate for the investigation of heterogeneous systems, due to lack of information on the spatial details. Conversely, local (i.e. space dependent) tallies such as the Feynman moments and the normalized variance are inherently useful in extracting information on the spatial behavior of the correlations, at the expense of an increased complexity.

Our main nding is that, in all tested congurations, the use of branchless collisions (as opposed to regular branching collisions) is very eective at quenching correlations and thus lowering the ratio between the apparent (ergodic) and real (ensemble) variance. In addition, while population control is generally advantageous in terms of weakening the eects of correlations, the impact of these methods is much milder than that of using branchless collisions. Using the last-event estimator in combination with sampling WOR was shown to be slightly less ecient than using branchless collisions for quenching spatial correlations.

In addition, we presented an exploratory work on the eects of branchless collisions in continuousenergy problems. Two dierent branchless sampling strategies were introduced : the reacting isotope is rst sampled as usual, and the branchless algorithm is applied afterwards with the corresponding microscopic cross sections ; or the reaction channel is sampled using macroscopic cross sections, and the reacting isotope is then sampled with probability laws depending on the chosen collision channel. Using branchless collisions on the material was found to better quench clustering in the system, although the improvement in terms of entropy and average surviving independent families was less signicant than in multi-group problems.

-Space-time correlations in time-dependent Monte Carlo simulations

In Chapter 5, we examined the correlations aecting the power iteration algorithm, in view of their relevance for the sampling of the equilibrium source for the time-dependent problem. In the following, we will focus on the space-time correlations aecting kinetic simulations, by relying on the insight gained in Part I of the thesis (Chapters 3 and 4), and in Chapter 5.

For the purpose of the analysis carried out here, we will relax most of the simplifying hypotheses introduced in the diusion-based analog transport models of Chapters 3 and 4. In particular, we will consider homogeneous congurations in a multi-group setting, similarly as in Chapter 5. Furthermore, we will use non-analog Monte Carlo methods, with several variance-reduction and population-control techniques : our main goal will be to assess their impact on correlations due to the interplay of these strategies.

Our investigation will be based on the methods and diagnostic tools developed in Chapter 5 for eigenvalue problems, with fairly minor modications. Some of the ndings related to power iteration, such as the emergence of spatial correlations leading to a patchy neutron density, will be shown to apply also to kinetic simulations. The role played by the ssion source distribution in the power iteration algorithm will be played by the neutron and precursor populations at the end of a time step in kinetic simulations. Similarly, observables related to generation-wise tallies in power iteration have a counterpart in tallies integrated over a time step in kinetic simulations. However, we will also illustrate some peculiar phenomena that are specic to kinetic simulations, without a direct counterpart in eigenvalue problems.

We will begin by adapting and extending the estimators that we have introduced in Chapter 5 for the correlations to the case of time-dependent problems. For this purpose, in Sec. 6.1 we will consider in particular the Shannon entropy and the Feynman moment of the neutron and precursor populations.

As a rst step towards the analysis of correlations in kinetic simulations, in Sec. 6.2 we will revisit the homogeneous slab reactor benchmark introduced by Nowak et al. in Ref. [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF], which we previously examined in Sec. 5.2 in the context of power iteration. For kinetic simulations, we will initially neglect precursors, we will model neutron transport as single-speed, and we will apply reective boundary conditions. In order to probe the sensitivity of our results to these simplifying assumptions, we will introduce several modications and assess their impact on correlations. To begin with, we will vary the reactor size, and we will also replace reective boundary conditions by leakage boundary conditions. The most relevant change will be the introduction of precursors : unsurprisingly, we will show that they deeply aect the amplitude and the spatial behaviour of the correlations.

Next, in Section 6.3, we will revisit the more realistic case of the multi-group homogeneous slabgeometry benchmark congurations introduced in Chapter 5. We will assess the correlations of the equilibrium neutron and precursor populations, as determined by running the power iteration described in the previous chapter, and how the equilibrium source aects the spatial correlations within the timedependent Monte Carlo simulations. Three dierent regimes will be explored : critical, subcritical and supercritical.

Conclusions for this chapter will be nally drawn in Sec. 6.4.

6.1 . Adapting the diagnostic tools to the time-dependent case

Based on the ndings of Chapter 5, we are now aware of the limitations of the global diagnostic tools for correlations. Nonetheless, such global observables still retain the advantage of being easy to interpret, and easy to compute. In order to facilitate the investigation of space-time correlations performed in the following sections, here we briey recall the denition of the main observables that we will use, namely the Shannon entropy and on Feynman moments. We refer the reader to Sec. 5.1.3 for a more thorough presentation.

. Neutron and precursor entropy

For time-dependent problems, we shall dene the Shannon entropy separately for neutrons and precursors, as the statistical behaviour of the two populations is expected to considerably dier. The Shannon entropy for neutrons reads

S n (t) = - B i=1 p n,i (t) log 2 (p n,i (t)), (6.1)
where p n,i is the fraction of the total neutron statistical weight in the spatial cell i at time t. Similarly, we dene the Shannon entropy for precursors as

S c (t) = - B i=1 p c,i (t) log 2 (p c,i (t)), (6.2)
where p c,i is the fraction of the total precursor statistical weight in the spatial cell i at time t. Although we may consider also the entropy of the total population, in realistic systems close to critical conditions the precursor population is much larger than the neutron population, so that the total entropy is entirely dominated by the precursor entropy. Therefore, we shall not include the total entropy in our investigation.

. Feynman moments

Contrary to Chapter 5, the question of ergodic vs. ensemble Feynman moment does not occur in pure time-dependent simulations. Nonetheless, it can still arise from the calculation of the equilibrium source using the power iteration. However, in the following we decided to simplify the study by looking at genuinely independent replicas, which are easier to interpret. By analogy with the time-independent case, we dene the time-dependent ensemble Feynman moment (or simply, Feynman moment) for an observable O as In this chapter, we apply the Feynman moments separately to the neutron and precursor spatial distribution in the equilibrium source (namely, Y S,n and Y S,c ), to the neutron and precursor spatial distribution during the kinetic simulation (namely, Y n and Y c ), and, when relevant, to the (neutron) collision counts (i.e. Y ψ ). Sometimes, for congurations where precursors are neglected, we will drop the n subscript.

Y O (r i , t) = V[O](r i , t) E[O](r i , t) , (6.3

. Analysis of a simplied homogeneous reactor benchmark

We begin our analysis by revisiting the simple benchmark problem proposed by Nowak et al. in Ref. 38 for the purpose of investigating particle clustering in power iteration, which we have previously introduced in Sec. 5.2. We consider a model of single-speed particle transport within a spatially homogeneous, reected slab of varying length L, with scattering cross section Σ s = 0.27 cm -1 , capture cross section Σ c = 0.002 cm -1 , ssion cross section Σ f = 0.002 cm -1 ; the ssion multiplicity ν p will be adjusted depending on the desired multiplication factor k e . The neutron speed is v = 10 3 cm • s -1 . Delayed neutron precursors are initially neglected.

For the purpose of running time-dependent Monte Carlo simulations, we will assume that the time interval [0, T ] is partitioned into K T = 100 equal time steps. We shall investigate a `large' system of length L = 400 cm and a `small' system of length L = 100 cm. By large system, we mean that L ≫ √ A,

i.e. the system size is large compared to the migration length ; conversely, in the small system we have L ∼ √ A. With the parameters described in the previous paragraph, we have √ A = 42.71 ± 0.02 cm for the large system, and √ A = 37.47 ± 0.01 cm . For spatial tallies, the slab is partitioned into B = 100 evenly spaced bins.

For a suitable choice of the ssion multiplicity ν p , the dominant eigenvalue of this conguration can be adjusted to any sought value ; the corresponding equilibrium distribution (i.e. the fundamental mode) will be spatially uniform over the slab, independently of ν p .

Neutron collisions are sampled using the branchless strategy, as described in Sec. 2.2.3. After collision events, neutrons undergo either Russian roulette, with threshold w R = 0.8 and survival weight w surv = 1, or splitting, with threshold w S = 2. At rst, no population control algorithm (such as combing) is applied at the end of the time steps. The neutron ights are sampled in an analog fashion.

In the following sections, this model will be made incrementally more complex by adding weight combing, leakage boundary conditions, and a single delayed neutron precursor family with an analog or forced-decay strategy, and we will assess the eect of each contribution on space-time correlations.

. Russian roulette and splitting as drivers of entropy steps

Our rst goal is to investigate the space-time correlations occurring in the time-dependent simulation of a critical, supercritical or subcritical conguration. We will characterize these correlations using the neutron and precursor Shannon entropy, and the Feynman moments. Occasionally, we will refer to other observables introduced in Chapter 5, such as the number of surviving neutron genealogical trees.

For this simple benchmark conguration, the equilibrium source can be sampled directly (the fundamental mode being spatially uniform), without resorting to the power iteration algorithm. The ideal Shannon entropy for the equilibrium source is therefore given by the expression given in Sec. 5.2, which is recalled here for convenience

S id = log 2 (N ) - B N B -1 B N N k=1 (B -1) -k N k k log 2 (k), (6.4)
where N is the number of particles (assuming unit statistical weights) and B is the number of spatial cells for the entropy tally.

About the absence of spatial correlations in the critical case

In the critical state, obtained by taking ν p = 2, the system presented in the previous section will lead to the same conclusions as in Section 5.2 : no spatial correlations arise and the neutron population behaves at any time like a collection of particles independently and identically distributed following a uniform spatial distribution in [-L/2, L/2] (recall that here the source particles are sampled independently). The neutron entropy will thus reach the ideal value given by Eq. 6.4. The reason for this behaviour is the same as in the case of power iteration : i) there is no loss of neutrons, and ii) the weight multiplier for branchless collision is of unity. Thus, the statistical weight of each individual neutron and the number of neutrons in the simulation will be constant. It is worth stressing that this behaviour is somewhat pathological, in the sense that it is due to the concurrence of several simplifying assumptions (branchless collisions ; lack of leakage ; homogeneous system ; criticality). If only one of these simplifying assumption is broken, non-trivial correlations will fatally develop.

Entropy steps in non-critical systems

More interesting situations arise when considering systems away from criticality. Non-stationary behaviour will be achieved by adjusting the ssion multiplicity at t = 0 ; the supercritical state is achieved by taking ν p = 2.5, and the subcritical state is achieved by taking ν p = 1.75. The time-evolution of the neutron entropy is plotted in Figs. [START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF].1 for subcritical and supercritical systems. The step-like timeevolution of these two observables can be explained by the interaction of the evolution of statistical weights with either Russian roulette or splitting. Let us rst analyze the supercritical case, and assume that we start with an initial population size N 0 = 250. In this case, the branchless weight multiplier is larger than unity, and the particle weight is strictly increasing with time ; on average, neutrons from the source will reach the splitting threshold at the same time, and consequently increase the number of particles in the system. In Fig. 6.1b, the splitting threshold will be reached on average at

τ th = ln w S ln f 1 vΣ t , (6.5)
where f is the branchless weight multiplier. For this one-speed supercritical system, we have τ th ≃ 0.6944 s, which roughly coincides with the time interval between successive entropy steps in Fig. 6.1b.

The number of Monte Carlo particles doubles upon reaching w S (we have set w S = 2), and the entropy correspondingly jumps to a new value. If the eect of the spatial correlations induced by splitting is negligible, the new value of the entropy coincides with the ideal Shannon entropy given by Eq. (6.4) and computed for N new = 2N 0 , as illustrated in Fig. 6.1b for the small system with L = 100 ; otherwise, when correlations have a stronger eect on the spatial distribution of neutrons, the entropy will temporarily converge towards a value smaller than the ideal entropy, as illustrated in Fig. 6.1b for the large system with L = 400. The subsequent entropy steps can be predicted in the same way, and the entropy converges to the asymptotic value S ∞ = log 2 (B) : for large time, spatial correlations are always quenched by the increasingly large population size.

In general, there will be particles that will reach the splitting (or roulette) threshold earlier or later than at time τ th . Therefore, as such occurrences accumulate over time, the amplitude of the entropy oscillations will be dampened over time as the occurrence of roulette and splitting become distributed more uniformly over time : asymptotically, such oscillations disappear (not shown here for conciseness). In multi-group and even more strikingly in continuous-energy systems, the multiple values of the branchless weight modier (which is energy-dependent) may prevent the observation of entropy oscillations.

Remark that in this system the statistics of family trees is trivial : neutrons cannot be destroyed, and the number of independent neutron trees is thus constant in time. This fact precisely illustrates that the analysis of the statistics of neutron genealogical trees may fail to catch the behaviour of spatial correlations. The neutron population in a non-analog Monte Carlo simulation may show spatial clustering even though xation is forbidden, although it may be arguable that the system is approaching xation because the number of families stays constant while the number of particles is increasing. However, we take this observation as a conrmation that the use of the statistics of family trees as a diagnostic tool for spatial correlations is questionable, as previously suggested by our analysis in Chapter 5.

A similar behaviour is detected in subcritical systems, as illustrated by Fig. 6.1a, this time driven by Russian roulette. The weight multiplier is now smaller than unity, and the statistical weight of neutrons is strictly decreasing. On average, neutrons reach the roulette threshold around the same time ; if the weight multiplier is suciently close to 1, on average f R = (w surv -w R )/w surv neutrons are eliminated, while the rest is restored to unit weight. The new entropy then converges to the ideal value corresponding to a number of particles N new = f R N 0 . In this case, the time step is again given by Eq. (6.5) with w S replaced by w R , leading to τ th ≃ 0.4458 s. The value of the entropy after each step is predicted by Eq. (6.4). Thus, in both subcritical and supercritical conditions, the Shannon entropy for this very simplied model is essentially driven by the total number of particles in the simulation, since all the particles have approximately the same weight (at least initially, for short times).

In the supercritical conguration, it is legitimate to ask whether the peculiar time-evolution of entropy leads to signicant spatial correlations. The slight dierences between the the entropy values for L = 100 and L = 400 in Fig. 6.1b faintly suggest so. To better characterize the eect of splitting on the spatial correlations, we rely on the Feynman moment of the neutron positions. The time evolution of the Feynman moment is plotted in Fig. 6.2. Figure 6.2a illustrates the absence of signicant spatial correlations when L = 100. To be more precise, around t = τ th ≃ 0.6944 s, the average neutron weight approaches w S for the rst time, and the value of the Feynman moment simply encodes this fact. There are no spatial correlations between neutrons, which is indicated by the fact that the Feynman moment is at.

At t = 0.8 s, most neutrons just underwent splitting, and their average weight is close to one. Spatial correlations are introduced by splitting, leading to an increase of the Feynman moment near the reective boundary conditions, which is reminiscent of the behaviour of the apparent-to-real variance ratio in the context of power iteration [START_REF] Sutton | Application of a discretized phase-space approach to the analysis of Monte Carlo uncertainties[END_REF]. However, these correlations are washed away between each splitting event, and the Feynman moment before the next entropy step (at t = 1.2 s, which is close to 2τ th ≃ 1.4 s) is systems, there is a small entropy dip after the rst time step, corresponding to the rst application of combing on an otherwise uncorrelated neutron population.

The analysis of Feynman moments in Fig. 6.5 shows a spatial shape at all times for both small and large systems, indicating the existence of spatial correlations. For the small system, the value of the Feynman moment is close to unity everywhere, suggesting that the eect of spatial correlations is in fact negligible. For the large system, however, the value of the Feynman moments signicantly diers from 1 ;

at this point, it is unclear whether it only encodes the average weight of the neutron population (recall that the Feynman moment is sensitive to the weight increase within a time step) or if it is an indicator of spatial correlations. The average weight after a single time step δt for a neutron of unit initial weight is given by w δt = f δt vΣt , leading to w δt ≃ 1.02. After combing, the average particle weight goes back to approximately 1. Therefore, the increase in weight during the time step is insucient to explain the value of the Feynman moment in Fig. 6.5b, which suggests the occurrence of signicant non-Poisson uctuations in the spatial distribution.

In the subcritical conguration (not shown here for conciseness), combing also suppresses the entropy steps induced by Russian roulette and leads to a convex entropy function. that the Feynman moments of the neutron positions (not plotted here for conciseness) are at, with a value dictated by the average weight of the neutron population. This is coherent with the fact that, in our implementation of combing, the length of the comb tooth is always very close to 1 (i.e. it is the average weight after combing), whereas the weight of neutrons before combing is always smaller than one (due to f < 1), and a single neutron cannot be selected twice by the combing procedure.

. The eect of vacuum boundary conditions on spatial correlations

If vacuum boundary conditions are used, the equilibrium source is not spatially uniform anymore, and must be sampled via the procedure described in Sec. 2.4.1 based on power iteration. The advantage of using combing in the power iteration is minimal : it increases the computation time without a signicant improvement in the quenching of spatial correlations ; therefore we simply use branchless collisions in the power iteration calculation. The size of the system now aects the multiplication factor k e , which complicates the comparison between systems of varying lengths : in order to ensure multiplication factors similar to the ones encountered in the previous example, the multiplication factor will be modied by adjusting the ssion multiplicity. The new parameters and multiplication factors are given by Tab. 6.1.

We also give the expected τ th for each system. We checked that 100 inactive generations were enough to reach statistical equilibrium in the preliminary power iteration calculation for each of these systems.

Analysis of correlations in the supercritical case

We rst focus on the supercritical case. The Feynman moments of the equilibrium source before and after combing are presented in Fig. 6.6. For both systems, the Feynman moment of the equilibrium source before combing is lower than one, which is directly related to the average weight of neutrons in the equilibrium source. In turn, this quantity is driven by the number of collisions in one generation of the power iteration, which is typically larger than the number of neutrons introduced in the power iteration (each neutron in the power iteration undergoes several collisions during one generation). After the equilibrium source has been sampled, it is re-normalized so that new statistical weight of each particle starting the kinetic simulation is given by

w ′ = N 0 W w, (6.6)
where W is the total weight of the equilibrium source before normalization, and w is the individual weight of the particle before normalization. The total weight of the equilibrium source is thus set to N 0 , which leads to an average weight signicantly smaller than 1. The fact that the shape of the Feynman moment does not really change after combing suggests that here combing is not eective in quenching spatial correlations. Only the value of the Feynman moment is changed. Given that the average neutron weight after combing is very close to 1, the value of the Feynman moments in Fig. 6.6 indicates that the spatial distribution of neutrons in the equilibrium source deviates from a locally Poisson distribution.

The spatial shape is close to the one of the Feynman moments of the ssion source and collision counts in Chapter 5, with two maxima near the vacuum boundaries.

The time evolution of the entropy is given by Fig. 6.7. In the simulations without combing, the initial entropy jump stems from the fact that the average weight of neutrons in the equilibrium source is considerably smaller than the Russian roulette threshold, as remarked in the previous paragraph, and a signicant number of Monte Carlo particles are immediately killed upon starting the time-dependent simulation ; the expected entropy of the neutron positions is consequently lower. For the small system, entropy oscillations occur. This is explained by the large leakage rate of this system (see Tab. 6.1) :

leakage signicantly reduces the number of neutrons in the system, thus reducing the expected entropy.

After time τ th ≃ 0.29 s, surviving neutrons reach the splitting threshold : the number of Monte Carlo particles increases, and so does the entropy. Since the system is supercritical, the average number of Monte Carlo particles created by splitting over a time step is larger than the average number of particles lost by leakage, and the entropy will globally increase. As time goes on, the amplitude of oscillations decreases. For the large system, the rate of leakage is small enough that it does not signicantly alter the entropy, and the system behaves similarly as the case with reective boundary conditions ; however, it is no longer possible to use Eq. (6.4) to interpret the values of the entropy at the intermediate steps, since the number of surviving particles is not known. Applying combing to the system eectively suppresses the initial jump, the entropy steps, and the entropy oscillations : the number of Monte Carlo particles closely follows the total neutron weight, and so does the entropy. Figure 6.8 shows the Feynman moment of the neutron positions for the small supercritical system.

The spatial shape of the Feynman moment without combing (see Fig. 6.8a) inherited from the shape of the Feynman moment for the equilibrium source is quickly forgotten, as illustrated by the curve for t = 0.24 s, which corresponds approximately to the time of the rst entropy oscillation. The spatial Feynman moment is at, indicating the absence of strong spatial correlations. At t = 0.36 s, the rst oscillation has just nished, and the Feynman moment now takes a non-uniform spatial shape, indicating that spatial correlations were introduced by splitting. The nal time illustrates how spatial correlations grow over successive entropy oscillations. For longer times, the Feynman moment of the neutron positions then takes a single maximum at the centre of the system. The value of the Feynman moment is related to the oscillations of the average neutron weight (and thus to the extrema of the entropy), and also increases with time due to the build-up of spatial correlations. When combing is applied (see Fig. 6.8b) the spatial shape inherited from the power iteration calculation of the equilibrium source survives longer, for the small system, although for larger times the spatial shape of the Feynman moment has again a single maximum at the centre of the system. The expected weight after a time step here is still w δt ≃ 1.02, as in Sec. 6.2.2, which is smaller than the value of the Feynman moment and thus indicates slightly non-Poisson uctuations in the spatial distribution of neutrons. In addition, the analysis of the Feynman moments clearly illustrates that the spatial correlations of the ssion source in power iteration (see e.g. Fig. 5.8a) are qualitatively dierent from the spatial correlations arising from the time-dependent simulations.

Figure 6.9 presents the Feynman moments for the large system with L = 400. The analysis of this gure raises some questions. First, it appears that the spatial shape of Feynman moment of the equilibrium source deeply inuences the shape of the Feynman moment of the neutron positions. Contrary to the previous paragraph, the Feynman moment takes two maxima near the boundaries of the system, for all times. This behaviour occurs both in the case with combing and the case without combing. Note that the time evolution of entropy, especially when applying combing, is insucient to discriminate between the case of the small system, whose spatial correlations have a peak at the centre, and the large system, whose spatial correlations have a peak near the boundaries.

Analysis of correlations in the subcritical case

The time evolution of the entropy for the subcritical case is given by Fig. 6.10. We observe again entropy oscillations, which evolve as described in the supercritical case, with dierent τ th due to the dierent physical parameters. The system is however subcritical : the number of Monte Carlo particles killed by Russian roulette or leakage exceeds the number of Monte Carlo particles created by splitting ; the total number of Monte Carlo particles will on average decrease with time, and so will entropy. Applying combing again removes the initial entropy jump and the oscillations. Analysis of the Feynman moments for these subcritical systems (not shown here for conciseness) leads to the following conclusions : for the small system, we have f > 1, thus the behaviour of the Feynman moment follows closely the one of the supercritical conguration investigated before ; for the large system, f < 1 and leakage is small, so that the behaviour of the Feynman moment is close to the one in Fig. 6.3 ; however, the spatial shape inherited from the equilibrium source survives for a time of the order of the mixing time τ p,D .

Analysis of correlations in the critical case

To conclude, the critical case formally corresponds to a situation with f > 1 and the increase of neutron weight exactly compensates the loss by leakage. Figure 6.11 shows the neutron entropy in the small and large systems. For the large system, given that τ th ≃ 9.4834s, we had to extend the time neutrons is very small before combing, and at with unit value after combing. Due to the ratio between neutrons and precursors, the total weight for precursor is much larger than for neutrons ; when normalizing the equilibrium source obtained at the end of the power iteration (as described in Sec. 2.4.1), neutrons are therefore assigned much smaller weights than precursors, leading to small values of the Feynman moment. This explains our rst comment. Moreover, the number of neutrons in the equilibrium source is much larger than the number of neutrons targeted by combing. This means that the spatial correlations induced by power iteration are carried only by the precursor population, and will typically take a time of the order of t ∼ 1/λ to relax, as it will be illustrated in the following paragraphs. This explains our second comment. Note that this behaviour should qualitatively carry over to Monte Carlo simulations with realistic parameters. In addition, the comments on the equilibrium source apply straightforwardly to the subcritical and critical systems.

It may seem wasteful to sample a large number of neutrons for the equilibrium source and throw away most of them in the combing phase. Actually, it is possible to control the number of particles in the equilibrium source before combing by dening a probability p acc to accept a collision as a valid sampling event of the equilibrium source. Upon a collision, particles for the equilibrium source are sampled the usual way with probability p acc , or the collision is ignored with complementary probability. To keep the sampling of the equilibrium source unbiased, the statistical weight w of any particle sampled following this procedure is divided by p acc . The benet of this algorithm is that it gives control of the memory footprint of the equilibrium source, and prevents sampling particles only to discard them immediately afterwards. On the other hand, this procedure may have two negative eects. First, p acc ≪ 1 may lead to very large weights. Second, if p acc is such that the number of particles in the equilibrium source before combing is smaller than the number of particles targeted by combing, then the combing procedure may introduce additional spatial correlations. This is illustrated in particular for the neutron population in Fig. 6.13 for p acc = 10 -3 . Note that in our context (i.e. the dynamics of neutrons is much faster than the dynamics of precursors), spatial correlations within the neutron population stemming from the sampling of the equilibrium source quickly disappear during the time-dependent process, as illustrated in the following section.

Analysis of correlations in the supercritical case

The time evolution of the entropy of the neutron and precursor populations in the supercritical case is plotted in Fig. 6.14. The neutron entropy (in Fig. 6.14a) is unaected by the decay strategy (forced or analog). The initial jump in the neutron entropy translates the fact that the neutron population is initially

Poisson distributed (as mentioned in the previous paragraph), but some correlations quickly grow in the initial neutron ssion chains. The linear behaviour occurring after the rst few seconds corresponds to the exponential increase of the neutron population, due to the system being supercritical. This interpretation is supported by the analysis of the Feynman moments for neutrons in Fig. 6.15a and 6.15b : after a few seconds, the Feynman moment of neutrons saturates at an average value larger than one. To decide whether this is an indicator of spatial clustering, we need to assess the expected weight of a neutron.

Within a time step, a neutron can increase its weight only through collisions ; if it is transformed in a behaviour is qualitatively the same as for the supercritical case, because the weight multiplier f > 1 : the value of the Feynman moment increases until it saturates around w δt ≃ 1.49, i.e. the expected weight of a neutron of unit initial weight at the end of a time step. On the other hand, in the large system (see Fig. 6.19b), we have f < 1 and the Feynman moment for neutrons is at. The value of the Feynman moment will decrease until it saturates at around w δt = 0.90.

The analysis of the Feynman moment of the precursor population is illustrated by Fig. 6.20 and Fig. 6.21. When using forced decay, the Feynman moment of the precursor population decreases with time in both systems, to a value that is much smaller than the expected precursor weight at the end of a time step (which, assuming unit initial weight, is w δt = 0.995). This is to be related to Fig. 6.17, and is similarly unexplained. When using analog decay, the Feynman moment becomes at.

Analysis of correlations in the critical case

Finally, the time evolution of the entropy in the critical system is shown in Fig. 6.22. Again, the neutron entropy has an initial jump, corresponding to the equilibrium source after combing being Poisson distributed (as illustrated in Sec. 6.2.4), while the neutron distribution during the time-dependent process is not. Once the jump ends, the entropy is constant for forced decay, and slightly decreasing for the analog decay, suggesting that the latter has a weak tendency to spatial clustering. This can be explained by the fact that forced decay leads to the production of more neutrons, which makes the neutron distribution

. Analysis of space-time correlations in multi-group systems

Relying on the insights regarding the eect of non-analog methods and precursors on the space-time correlations of the time-dependent simulations obtained for the simple system in Sec. 6.2, we shall now revisit the slab benchmarks introduced in Chapter 5. Due to the large number of possible permutations of variance-reduction techniques, simulation parameters and benchmark congurations, and in view of the fact that each simulation involves an extremely high computational cost, we decided to restrict our analysis to homogeneous media, with a single set of variance-reduction techniques and space-time discretization.

The equilibrium source for the systems presented in Sec. 6.3.1 needs to be computed using a preliminary power iteration calculation, and we choose to use branchless collisions without combing, as detailed in Chapter 5. In addition, in order to limit the memory occupation, we use the alternative sampling procedure presented in Sec. 6.2.4, with an acceptance probability p acc = 10 -3 . The time-dependent calculations adopt branchless collisions for neutrons ; a single comb is applied on the total population at each time step, and on the equilibrium source. For precursors, we use the combined precursor particle (presented in Sec. 2.4.2), and we apply forced decay. If a neutron is produced through forced decay with a statistical weight w < w R , it is immediately tested for Russian roulette (i.e. we do not wait for the rst Parameters g = 1 (fast) g = 2 (epithermal) g = 3 (thermal) ξ p (g) [-] 0.878198 0.121802 0 Σ c,g (super) 9.300000 × 10 -4

1.490355 × 10 -2 6.906277 × 10 -2 Σ c,g (sub)

9.460000 × 10 -4 1.510355 × 10 -2 6.919277 × 10 -2 Table 6.4 : Capture cross sections for the reactivity insertions in system 1, along with the new prompt fission spectrum. The energy groups are indexed by g.

Parameters

g = 1 (fast) g = 2 (epithermal) g = 3 (thermal) Σ c,g (super) 9.302099 × 10 -4

1.493439 × 10 -2 7.0600812 × 10 -2 Σ c,g (sub)

9.4820991 × 10 -4 1.5134394 × 10 -2 7.0720812 × 10 -2 Table 6.5 : Capture cross sections for the reactivity insertions in system 2. The energy groups are indexed by g.

. Investigating the spatial correlations in the equilibrium source

We start our investigation by considering the kinetic source for system 1 and 2. Based on our previous investigations in Sec. 6.2.4, we expect spatial correlations stemming from the equilibrium source to be carried mostly by the precursor population. The Feynman moments of the neutron and precursor populations after combing are shown in Fig. 6.25 for system 1 and in Fig. 6.26 for system 2. For the sake of simplicity, in the following we refer to them as `neutron source' and `precursor source'.

As suggested in Sec. 6.2.4, the choice of the acceptance probability leads to uctuations in the Feynman moment for the neutron source, indicated by the noisy behaviour of the Feynman moment of the (thermal) neutron source in Figs. [START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF].25a and 6.26a. Spatial correlations are induced by combing, which manifests itself through the convex shape similar to the one observed in Fig. 6.13.

The behaviour of the Feynman moments of the precursor source is induced by the use of the combined precursor particle ; the value of the Feynman moment for each family is driven by the fractional weight for family j, which (for a particle stemming from the equilibrium source) is given by

w eq j = λ λ j β j β , (6.7)
with denitions as in Sec. 2.4.1. This distribution is a consequence of Eq. (2.34). The expected weight after combing is very close to 1. Thus, the expected fractional weight for each precursor family in the equilibrium source is only driven by the delayed nuclear data in Tab. 6.3 ; numerical values are given in Tab. 6.7. The values of the Feynman moments are proportional to the values of the expected fractional weights. However, the absolute values of the Feynman moments are dierent from the expected fractional weights, which indicates deviations from a locally Poisson distribution. Even in situations where the dominance ratio is large (for system 2, we recall that the dominance ratio is R = 0.9957), the use of branchless collisions in the power iteration eectively suppresses the spatial correlations that usually arise from the sampling of the kinetic source. This is revealed by the absence of spatial shape for the Feynman moment of the precursor populations (except close to the boundaries). Finally, because of the use of the combined precursor particle, the shape of the spatial correlations in the precursor source is independent of the precursor family.

. Analysis of space-time correlations in the critical case

The Shannon entropy for the neutron and precursor populations is represented in Fig. 6.27. The neutron entropy follows the same trend as in the simplied systems investigated in Sec. 6.2.4 : the initial jump happens on a very short timescale, suggesting that the spatial correlations within the neutron source are almost instantaneously forgotten. The long-time behaviour of the neutron entropy in system 1 appears to converge to a constant value, whereas the neutron entropy in system 2 seems to decrease, suggesting a build-up of spatial correlations within the neutron population. However, the decrease could also be related to the value of the multiplication factor, which is slightly subcritical, and additional investigations relying on the Feynman moments will be necessary. In system 1, the precursor entropy appears to reach saturation for large times, whereas it reaches a maximum near t ≃ 8 s for system 2, before decreasing.

This last remark could be either an indicator of spatial clustering within the precursor population, or the consequence of the slight subcriticality of system 2.

As for the Feynman moments, we rst focus on the space-time correlations within the neutron population. For neutrons, whose population is small at any given time, the Feynman moment of the neutron positions is typically noisy, and a satisfactory statistical accuracy could not be achieved. This makes the interpretation of the Feynman moments more dicult : for illustration, Fig. 6.28 represents the Feynman moment of the position of neutrons in the thermal group at dierent times (the Feynman moments for the epithermal and fast group were too noisy). Figure 6.28 suggests that the Feynman moment saturates almost instantaneously in both systems. Note that with the current parameters, τ th ≃ 2.215 × 10 -4 s for the thermal group, which is much smaller than a single time step. It is therefore dicult to compare the value of the Feynman moment to the expected neutron weight ; nonetheless, the existence of spatial correlations within the neutron population are suggested by the large value of the Feynman moment. A clearer picture can be obtained from the analysis of the Feynman moments of the collision counts, which are integrated over a time step, as illustrated in Fig. 6.29 : after a short time (less than 0.1 s), the Feynman moments saturate for each group, and are constant for long times, suggesting that the critical catastrophe is successfully averted.

As suggested in Sec. 6.2.4, we expect that long-term correlations arise within the precursor population. This is illustrated by Fig. 6.30 for the large system, at t = 0.1 s (i.e. when spatial correlations are still aected by the kinetic source), and at t = T = 10 s (i.e. the nal simulation time). Figure 6.30 clearly illustrates that at t = T not all the precursor families have forgotten the source correlations and relaxed to the time-dependent correlations : the families with the slowest decay (i.e. 1/λ 1 ≫ T ) still exhibit the (almost) at shape inherited from the sampling of the kinetic source, while the spatial correlations within families with faster decay are now peaked in the centre of the system. For families with faster decay, the Feynman moment signicantly increases with time : at much longer times, we expect this remark to extend also to the families with slower decay. The same conclusions would apply to system 1, which we do not detail here for the sake of conciseness.

. Analysis of space-time correlations with reactivity insertions and withdrawals

We now consider the behavior of the clustering indicators in scenarios where the system is prepared on the critical state, but an amount of reactivity is inserted or withdrawn at t = 0. The reactivity changes are given in Tab. 6.6.

The corresponding Shannon entropy for the neutron and precursor populations is plotted in Fig. 6.31.

In both cases, the neutron entropy undergoes a prompt jump at short times, which is again explained by the fact that spatial correlations within the kinetic source are somewhat dierent from those of the timedependent process. In the supercritical case (i.e. reactivity insertion), the height of the jump is smaller than in the critical or subcritical case (i.e. reactivity withdrawal). After the prompt jump, the neutron the saturation at large times of the Feynman moments in the critical conguration, this suggests that, in the supercritical case, the Feynman moment might also be independent of time. This reasoning should also hold true for the subcritical case for intermediate times, i.e. as long as the population size does not become too small. The Feynman moment in the subcritical case will eventually recover a time dependency for large times, as the population size vanishes. For the sake of conciseness, in Fig. 6.32 we only provide the curves for system 2, but the behaviour of system 1 is essentially the same.

To conclude, we present the Feynman moments of the precursor populations. The comparison of Figs. 6.33 and 6.34 for the non-stationary congurations of system 2 with respect to Fig. 6.30 for the stationary conguration indicates that spatial correlations grow similarly. This suggests that reactivity insertions and withdrawals only marginally aect the spatial correlations within the precursor population.

The same conclusions hold true for system 1, which is not presented here.

(a) Case of system 1.

(b) Case of system 2. Blue is for t = 0.001 s ; red is for t = 0.1 s ; orange is for t = 1 s, and green for t = 10 s.

. Conclusions

In this chapter, we investigated the eects of variance-reduction methods on the space-time correlations in time-dependent Monte Carlo simulations. Based on our discussion in Chapter 5 on estimators for spatial correlations, we focused on the analysis of the Shannon entropy (for an easy to compute, global observable), and of Feynman moments (for a local observable, leading to a more detailed analysis of spatial correlations). The impact of non-analog methods was rst assessed in a simplied benchmark system, allowing for easier interpretation. We then revisited in a time-dependent setting the spatially homogeneous benchmark problems introduced in Chapter 5.

Based on a simple one-speed transport problem without precursors, in a slab with reective boundary conditions, we showed that the combination of Russian roulette or splitting with branchless collisions lead to the occurrence of regularly spaced `steps' in the time evolution of the Shannon entropy. The amplitude of the steps was shown to decrease with time. In supercritical systems, where splitting may occur, we showed how spatial correlations grow over successive entropy steps, and we discussed the conditions dictating whether signicant spatial correlations will emerge. In subcritical systems, the entropy steps had no inuence on spatial correlations. We have also shown that the use of weight combing suppresses the entropy steps, regardless of the reactivity of the system. In supercritical system, combing introduces additional spatial correlations ; in subcritical system, we showed that combing was again not eective in quenching possible spatial correlations. Replacing reective by leakage boundary conditions led to a shape of spatial correlations within the equilibrium source (computed using power iteration) that was qualitatively dierent from the shape of spatial correlations arising in the time-dependent process. We showed that spatial correlations within the equilibrium source may have a long-lasting eect on the spatial correlations of the time-dependent simulations. We also illustrated that the interaction of splitting with leakage boundary conditions leads to oscillations in the neutron entropy, which are also related to the build-up of spatial correlations. The introduction of `realistic' precursors shows that spatial correlations within the equilibrium source carry over in the time-dependent simulation through precursors. The key ndings of the case without precursors extend to the case with precursors.

Finally, we investigated the space-time correlations emerging in the spatially homogeneous multigroup transport benchmarks introduced in Chapter 5. In this case, precursors were taken into account.

Relying on the results obtained in Sec. 6.2, we characterized the evolution in time and in space of the correlations. Our ndings suggest that the eect of reactivity insertions or withdrawals on spatial tional sampling strategies and implement unusual scores, which is cumbersome due to the complexity of production-level codes. The rst Monte Carlo mini-app, named JOFFREY, is devoted to analog (i.e., without variance-reduction techniques) time-dependent transport simulations using the diusion approximation, where particle displacements are sampled using Brownian motion, in one-dimensional geometries.

The second Monte Carlo mini-app, named COYOTE, handles more realistic systems in a multi-group spatially heterogeneous framework, including power iteration to sample the stationary (critical) state of neutrons and precursors, and time-dependent transport.

In the rst part of the thesis (Chapters 3 and 4), we have examined the behaviour of uctuations and correlations in time-dependent Monte Carlo simulations for the simpler case where sampling is analog (which means that no variance-reduction techniques are used) and the diusion approximation is used.

These choices were made in order to set a framework amenable to exact solutions for the moment equations describing the two-point correlation functions, against which Monte Carlo simulations can be compared.

For this purpose, in Chapter 3 we have introduced a stochastic model where a collection of particles undergo diusive displacements obeying a Brownian motion, coupled with a Galton-Watson birth-anddeath process that mimics collision events where particles can be captured (and thus disappear) or ssion (and thus give rise to a random number of descendants). The system is one-dimensional, with mass-preserving reecting boundary conditions. Precursors have been modelled using an exponentially distributed delay between the ssion events and the emission of further particles in the system at the collision sites. Despite some extremely strong simplications, this model preserves the key ingredients of time-dependent particle transport in multiplying media, and provides thus meaningful insights on the behaviour of uctuations and correlations in analog Monte Carlo simulations. Within this framework, using the Pál-Bell backward formalism we have derived the moment equations, which yield in particular the average particle number and the two-point correlation function. We have analytically solved these equations, and exact results have been compared to analog Monte Carlo simulations, used as a reference.

Special emphasis has been given to the critical case, where births by ssion are compensated by deaths by capture, which is crucial for reactor control. We identied the typical timescales over which correlations evolve : the time necessary for the prompt correlation dynamics to relax, the time necessary for correlations to achieve their asymptotic spatial shape, and the time required to observe catastrophic uctuations of the total population size. The results obtained in the rst part of the thesis provide a sound theoretical framework for the investigation of kinetic Monte Carlo simulations, but come with several restrictions : they apply only to the case of analog sampling, without any variance-reduction method, and to single-speed diusion in spatially homogeneous systems. Population control can be taken into account, but only through highly simplied models where the population size is kept exactly constant. Practically all production Monte Carlo codes use some forms of non-analog sampling, involving variance-reduction and population-control methods : understanding the impact of such techniques on the behaviour of space-time correlations is thus crucial in order to increase the reliability of kinetic simulations.

To overcome the limitations of the models used in Chapters 3 and 4, in the second part of the thesis (Chapters 5 and 6) we have therefore examined the behaviour of uctuations and correlations in a more realistic setting. The main generalizations have consisted in switching to fully non-analog Monte Carlo sampling, and in relaxing the diusion approximation by considering multi-group transport processes.

In Chapter 5 we have investigated the impact of non-analog collision sampling, variance-reduction and population-control methods on the correlations that occur in the Monte Carlo implementation of the power iteration algorithm. Power iteration is the standard approach used to estimate the fundamental mode of the k-eigenvalue formulation of the Boltzmann equation, which are key to the sampling of the critical source, i.e., the equilibrium reactor conguration, for kinetic simulations. Our investigation has been carried out on a set of simple, yet meaningful benchmark problems encompassing homogeneous and heterogeneous geometries. Among the probed non-analog Monte Carlo methods, we have in particular focused on branchless collisions, which force the emergence of a single neutron at each collision and thus quench the correlations due to `branches' in particle histories coming from common ancestors.

Furthermore, we have focused on combing and sampling WOR (without replacement) as populationcontrol techniques.

For the purpose of detecting the presence of strong correlations and assessing the relative benets of various non-analog Monte Carlo methods in reducing their impact, we have selected several distinct tallies : the Shannon entropy, the average pair distance squared, the average number of surviving families, the Feynman moments and the normalized variance. We have shown that all the proposed tallies are useful to detect the presence of anomalous uctuations in the examined systems. However, global tallies such as the entropy and the average square pair distance, while being easy to use and interpret, have been shown to be inadequate for the investigation of heterogeneous systems, due to lack of information on the spatial details. Conversely, local (i.e. space dependent) tallies such as the Feynman moments and the normalized variance are inherently useful in extracting information on the spatial behavior of the correlations, at the expense of an increased complexity.

Our main nding was that, in all tested congurations, the use of branchless collisions (as opposed to regular branching collisions) is very eective at quenching correlations. In addition, while population control is generally (but not always) advantageous in terms of weakening the eects of correlations, the impact of these methods is much milder than that of using branchless collisions.

After examining the case of power iteration, in Chapter 6 we have reverted to the case of timedependent Monte Carlo simulations. Based on the analysis of Chapter 5, we have primarily focused on the Shannon entropy and on Feynman moments. The eect of non-analog methods was rst characterized in a simplied system, allowing for easier interpretation. We have probed critical, sub-critical and supercritical congurations, with various combinations of system sizes and boundary conditions. Oscillations in the neutron entropy were observed and related to the build-up of spatial correlations, whose shape depends on the interplay between the reactivity of the system, the specic variance-reduction and populationcontrol method used, and the kind of boundary conditions. In some cases, the oscillations of the entropy function were shown to be persistent over long times.

Finally, we investigated the space-time correlations emerging in the more realistic homogeneous benchmarks introduced in Chapter 5. We detailed how spatial correlations grow over time. Our ndings suggest that the eect of reactivity insertions on spatial correlations is marginal, at least in the regime where reactivity is small. Notably, the critical catastrophe is eectively averted by the use of non-analog methods.

The work carried out in this thesis has shed some light on the behaviour of uctuations and correlations in time-dependent Monte Carlo simulations, starting from highly simplied settings and progressively making them more complex in order to broaden the domain of validity of our ndings. In view of getting a deeper grasp of the interplay between sampling methods and correlations, we have kept even our most sophisticated benchmark problems much simpler than realistic applications that could be met in production Monte Carlo codes. On one hand, this has allowed estimating and analyzing tallies that would be generally out of reach due to computational issues, such as two-point correlation functions. On the other hand, we are aware that the conclusions drawn in one-dimensional congurations using multi-group nuclear data might not be representative of the behaviour observed in three-dimensional congurations using continuous-energy nuclear data. Furthermore, it is important to stress that kinetic Monte Carlo methods are a relatively young, thriving and still unsettled research eld : novel variance-reduction or population control algorithms might appear in the near future, either conceived from scratch for timedependent problems or adapted from strategies already existing for stationary problems. Such changes might force us to reconsider the outcomes of the analysis presented here, especially for the case of kinetic simulations discussed in Chapter 6.

Several paths for future work are quite naturally prompted by the ndings of this thesis. As briey illustrated in Chapter 5, replacing multi-group with continuous-energy nuclear data might have a signicant impact on the behaviour of the variance-reduction techniques with respect to uctuations and correlations. A more thorough investigation of realistic three-dimensional benchmarks with continuousenergy nuclear data is thus mandatory, which would simultaneously call for the development of improved diagnostic tools to detect correlations, in order to handle the increased computational complexity and memory footprint.

Another hint concerns the analysis of the interplay between the spatial scale of correlations and the spatial scale of the Monte Carlo tallies. Based on the Monte Carlo simulation results presented in Chapters 5 and 6, it is apparent that the emergence of spatial correlations is intimately related to the competition between the linear size of the particle clusters and the one of the regions over which the tallies are estimated. A similar interplay might occur also with respect to the time scales : the one of correlations and the one of the tallies in kinetic simulations ; this conjecture should be probed numerically.

It would be also interesting to develop a theoretical framework enabling to adapt the tallying mesh in order to quench the eects of correlations.

Related to variance-reduction techniques, a further topic for future research would be the systematic exploration of zero-variance schemes, along the lines proposed in Ref. 100, to achieve a coherent framework for a time-dependent Consistent Adjoint-Driven Importance Sampling (CADIS) strategy. These methods would have the potential to become game-changers in kinetic Monte Carlo simulations, leading to considerable improvements in the gure of merit, similarly as what happens already for the case of stationary shielding problems, where CADIS is nowadays customarily employed.

Finally, although in this thesis for the sake of simplicity we have focused on kinetic simulations alone, in most applications the eects of physical feedback must be taken into account and deeply aects the evolution of the neutron and precursor population. If the overall feedback coecient is negative, it will basically act as an average-restoring mechanism, preventing large excursions around the mean value of the population density. Only the average values coming from the kinetic Monte Carlo simulations can be fed to the coupling codes, which are deterministic. Statistical uncertainty can be propagated by performing independent replicas, at the expense of an increased computational cost. In order to compensate for the extra computer time, one might be tempted to reduce the number of simulated histories per replica, but this would amplify the bias due to the fact of feeding a uctuating quantity to the coupled solver, the coupling being non-linear. Theoretical and practical issues make therefore these problems non-trivial : an ambitious goal for future exploration would be the extension of the general framework presented in this thesis to the case where kinetics is coupled to thermal-hydraulics. La densité de neutrons, c'est-à-dire le nombre moyen de particules dans un volume élémentaire de l'espace des phases, obéit à l'équation de Boltzmann [2]. Les solutions analytiques de cette équation sont généralement inatteignables, à l'exception du cas de certains systèmes simpliés, et l'usage de la simulation numérique est donc nécessaire. Grâce à la linéarité de l'équation de Boltzmann et à la dimension intermédiaire de l'espace des phases (qui n'est ni trop grande, ni trop petite), les méthodes Monte-Carlo et les méthodes déterministes sont toutes deux utilisables pour obtenir des solutions numériques, chacune de ces méthodes bénéciant d'avantages et d'inconvénients propres. L'équation de Boltzmann en conditions non stationnaires est traditionnellement résolue en utilisant les méthodes déterministes, dans lesquelles l'espace des phases est discrétisé. À cause du très grand nombre d'inconnues (de l'ordre de ∼ 10 14 ) qui résulterait d'une discrétisation susamment ne de l'espace, de la direction et du temps, la plupart des codes déterministes modernes adoptent une approche en deux temps, lors de laquelle le résultat d'une simulation nement discrétisée à l'échelle d'un assemblage est utilisé an de préparer les données nucléaires homogénéisées et condensées. Ces données sont ensuite injectées dans un solveur temporel simplié, dont la discrétisation énergétique est assez grossière, an d'eectuer une simulation à l'échelle du c÷ur complet [START_REF] Miao | Analysis of correlations and their impact on convergence rates in Monte Carlo eigenvalue simulations[END_REF]. De plus, il est courant de séparer la variable temporelle du reste des variables de l'espace des phases classique [START_REF] Dulla | The quasi-static method revisited[END_REF][START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF][START_REF] Dulla | Some new thoughts on the multipoint method for reactor physics applications[END_REF].

A -Résumé Détaillé en Français

Les solveurs déterministes sont rapides, mais cette rapidité vient au prix de biais dus à la discrétisation et à l'usage de modèles dépendants du type de réacteur (par exemple, l'auto-protection). Ces biais impliquent que cette classe de méthodes numériques doit être validée par comparaison avec des méthodes de simulation de haute-délité ou bien avec des résultats expérimentaux [START_REF]Safety Margins of Operating Reactors -Analysis of Uncertainties and Implications for Decision Making[END_REF]. Les données expérimentales sur les réacteurs en régime stationnaire sont généralement plutôt accessibles, bien que leur appartenance aux opérateurs exploitants des centrales nucléaires limitent cette accessibilité. Pour cette dernière raison, plusieurs initiatives ont récemment été promues an d'établir des résultats de référence internationaux pour la validation des codes de calcul (y compris en incluant la déplétion du combustible et les couplages multi-physiques). On peut citer VERA (Virtual Environment for Reactor Applications) 4 . En revanche, dans les régimes non stationnaires tels que les tran- sitoires opérationnels résultant de l'insertion des barres de contrôle ou bien les campagnes expérimentales destinées à évaluer le comportement du réacteur lors de sollicitations extérieures, la situation est bien plus problématique. De plus, les cas extrêmes, tels que les transitoires accidentels pouvant mener à la dégradation du combustible, sont dangereux et ne peuvent être explorés par les moyens expérimentaux, malgré leur importance pour la validation des codes de simulation numérique [START_REF]Research reactor benchmarking database: Facility specication and experimental data[END_REF]. Dans le cadre de ces considérations, les simulations Monte-Carlo tiennent une place à part, car elles ne sont pas sujettes aux erreurs de discrétisation [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF] et peuvent ainsi jouer le rôle de référence pour la validation des solveurs déterministes ; de plus, elles sont de fait inévitables pour évaluer les biais déterministes dans les régimes non stationnaires.

A.1.1 . Simulations Monte-Carlo : du stationnaire au transitoire Les simulations Monte-Carlo pour le transport de particules se basent sur l'échantillonnage aléatoire des trajectoires des particules dans l'espace des phases. Pour les applications intéressant la physique des réacteurs, les particules sont principalement des neutrons, des précurseurs de neutrons retardés, et des photons. Dans le cadre de cette thèse, nous nous concentrerons sur les neutrons et les précurseurs. Leurs interactions aléatoires avec le milieu environnant sont échantillonnées en accord avec les lois de probabilités données dans les librairies de données nucléaires ; le suivi des particules est eectué en calculant l'intersection des vols entre chaque collision avec les cellules constituants le système géométrique. Les trajectoires sont suivies jusqu'à ce que les particules disparaissent, par absorption ou bien en s'échappant des limites du système. Le long de chaque trajectoire, des observables physiques telles que les taux de réactions sont estimées en sélectionnant adéquatement des scores dont on calcule la moyenne et la variance. Grâce à la loi des grands nombres, la moyenne sur l'échantillon donne une estimation non biaisée de l'observable physique associée ; la variance de l'échantillon donne quant à elle une estimation de l'incertitude statistique sur la moyenne de l'échantillon, grâce au théorème central limite [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. La variance de la moyenne de l'échantillon est inversement proportionnelle au nombre d'histoires échantillonnées, ce qui implique que les simulations Monte-Carlo sont généralement bien plus longues que les simulations déterministes. Cependant, cette limitation est compensée par l'absence d'approximations dans résultats temps correspond souvent à la condition d'équilibre (ou stationnaire) du réacteur, ce qui signie que les particules commençant la simulation cinétique héritent des corrélations aectant le calcul aux valeurs propres nécessaire pour échantillonner la conguration à l'équilibre des particules [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. Quand le nombre de particules est petit, l'eet de ces corrélations est amplié, et se manifeste sous la forme de regroupements spontanés de particules (`clustering'), aboutissant à des distributions spatiales irrégulières [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF] telles qu'illustré sur la Figure A.1. De plus, les simulations Monte-Carlo cinétiques sont aectées par des problèmes qui leur sont plus spéciques : les histoires peuvent `brancher' à cause des évènements de ssion conduisant à l'apparition de neutrons secondaires, ce qui augmente la variance et se trouve être à l'origine des corrélations spatio-temporelles, même lorsque le réacteur est critique (stationnaire). Dans certains cas, ces eets peuvent mener à une croissance incontrôlée de la variance en fonction du temps, appelée la catastrophe critique [START_REF] De Mulatier | The critical catastrophe revisited[END_REF].

Dans ce qui suit, nous donnons un rapide tour d'horizon des connaissances actuelles sur l'eet des corrélations dans les simulations Monte-Carlo, ainsi que les questions encore ouvertes et leur importance pour diérentes applications.

Corrélations dans les calculs aux valeurs propres Monte-Carlo

Comme illustré dans cette thèse, la distribution à l'équilibre des neutrons et des précurseurs peut être déterminée par des méthodes Monte-Carlo simulant les chaines de ssion sur plusieurs générations (une génération étant dénie comme l'histoire d'un neutron depuis sa naissance jusqu'à sa mort par fuite ou par absorption), jusqu'à ce que le régime stationnaire soit atteint [START_REF] Lux | Monte Carlo particle transport methods: neutron and photon calculations[END_REF]. À chaque génération, la généalogie de n'importe quel neutron peut être retracée jusqu'à un ancêtre, et l'ensemble des neutrons partageant un même ancêtre commun forme une famille : des évènements issus d'une même famille seront corrélés. Ces corrélations vont aecter à la fois le comportement statistique d'un score au sein d'une même cellule spatiale au cours de générations successives, et le comportement des scores dans diérentes cellules spatiales à une génération donnée : ces deux points de vue sont intimement reliés. L'évaluation des corrélations générationnelles est un problème de longue date. Dans les calculs aux valeurs propres Monte-Carlo, la moyenne et la variance sont typiquement estimées au cours de générations successives, en faisant l'hypothèse que les scores collectés lors de chaque génération sont indépendamment et identiquement distribués. Cela correspond à une moyenne ergodique, par analogie avec l'hypothèse ergodique en physique statistique à l'équilibre. Bien que les scores soient eectivement identiquement distribués (une fois le système convergé vers l'état d'équilibre), ils sont aussi corrélés, comme discuté précédemment, et cela aecte l'estimation de l'incertitude sur leur valeur moyenne.

Brissenden et Garlick ont les premiers remarqué que négliger les corrélations générationnelles en estimant la variance au moyen de la moyenne ergodique menait à une sous-estimation de la variance des scores [START_REF] Brissenden | Biases in the estimation of k e and its error by Monte Carlo methods[END_REF] : pour cette raison, il est nécessaire de faire la distinction entre la variance apparente (sous-estimée) et la vraie variance obtenue par moyenne d'ensemble sur un ensemble de répliques totalement indépendantes 5 . Ueki et al. [START_REF] Ueki | Autocorrelation and dominance ratio in Monte Carlo criticality calculations[END_REF] ont montré que la diérence entre la variance apparente et la vraie variance peut-être considérable pour des scores dépendant de l'espace. Ils ont aussi montré que le taux de convergence en fonction du nombre de particules simulées dière de celui attendu pour des histoires indépendantes, bien que Herman ait montré plus tard que le comportement attendu est retrouvé asymptotiquement pour des populations de grande taille [START_REF] Herman | Monte Carlo and thermal hydraulic coupling using low-order nonlinear diusion acceleration[END_REF]. Min-Jae et al. ont découvert que le ratio de la variance apparente sur la vraie variance n'était pas uniforme dans l'espace [START_REF] Lee | Multigroup monte carlo reactor calculation with coarse mesh nite dierence formulation for real variance reduction[END_REF].

Plusieurs méthodes visant à prédire la vraie variance en partant de la variance apparente ont été proposées, le plus souvent avec des hypothèses simplicatrices : Yamamoto et al. ont modélisé la sousestimation de la variance des scores locaux (ou dépendant de l'espace) ainsi que sa dépendance sur la taille des cellules spatiales [START_REF] Yamamoto | Prediction on underestimation of variance for ssion rate distribution in monte-carlo calculation[END_REF]. Sutton a appliqué une approche de discrétisation de l'espace des phases an de comprendre l'hétérogénéité du ratio de la variance apparente sur la vraie variance [START_REF] Sutton | Application of a discretized phase-space approach to the analysis of Monte Carlo uncertainties[END_REF]. Miao L'impact des corrélations spatiales dans les calculs aux valeurs propres Monte-Carlo a été récemment remarqué par Dumonteil et al., qui les premiers ont noté l'émergence des regroupements de neutrons [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF] ; des outils de diagnostic ont été développés plus tard par Nowak et al. [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF] en approfondissant des recherches précédentes [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. Bien que ces corrélations spatiales n'aectent pas aecter la moyenne des scores Monte-Carlo en eux-mêmes, Cosgrove et al. ont montré qu'elle pourrait avoir une inuence lorsque des simulations Monte-Carlo sont couplées à un solveur pour la déplétion, et ainsi amplier des instabilités numériques préexistantes [START_REF] Isotalo | Preventing xenon oscillations in Monte Carlo burnup calculations by enforcing equilibrium xenon distribution[END_REF][START_REF] Dufek | Numerical stability of the predictorcorrector method in Monte Carlo burnup calculations of critical reactors[END_REF], aboutissant ainsi à l'apparition d'un biais sur la moyenne des observables [START_REF] Cosgrove | Neutron clustering as a driver of Monte Carlo burn-up instability[END_REF]. Sutton a proposé d'étudier le comportement spatial des neutrons au moyen du comportement statistique des arbres généalogiques de neutrons, en notant que les corrélations étaient ampliées lorsqu'il ne restait qu'un petit nombre de familles indépendantes (ou bien des familles plus grandes) dans la population de neutrons [START_REF] Sutton | Neutron clustering in Monte Carlo iterated-source calculations[END_REF][START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF]. Concernant la question du `clustering', les résultats présents dans la littérature sont assez contradictoires, et il ne semble pas exister de véritable consensus.

Certains auteurs ont suggéré que le `clustering' des neutrons pourraient avoir un impact sur la moyenne des scores, en plus d'aecter l'estimation de la variance [START_REF] Dumonteil | Clustering and traveling waves in the Monte Carlo criticality simulation of decoupled and conned media[END_REF][START_REF] Fröhlicher | Generational variance reduction in monte carlo criticality simulations as a way of mitigating unwanted correlations[END_REF]. Au contraire, d'autres ont suggéré que les corrélations spatiales n'avaient pas d'eets notables sur l'estimation des incertitudes statistiques [START_REF] Mickus | Does neutron clustering aect tally errors in Monte Carlo criticality calculations?[END_REF].

Corrélations dans les simulations Monte-Carlo cinétiques

Les simulations Monte-Carlo dépendant du temps ont été introduites assez récemment dans le contexte de la physique des réacteurs, ce qui explique que l'étude du comportement des corrélations dans cette classe de problèmes en soit encore à ses balbutiements. De plus, malgré des avancées signicatives concernant les méthodes de réductions de variance, le coût en calcul des problèmes cinétiques reste très élevé, et l'étude des corrélations est le plus souvent eectuée dans des modèles avec diérents niveaux de simplication.

Dumonteil et al. ont évalué les corrélations spatiales à un temps d'observation donné en négligeant la contribution des précurseurs et en supposant que le milieu était inni [START_REF] Dumonteil | Particle clustering in Monte Carlo criticality simulations[END_REF]. L'eet du milieu ni sur les corrélations spatio-temporelles a plus tard été pris en compte [START_REF] Zoia | Clustering of branching Brownian motions in conned geometries[END_REF][START_REF] Zoia | Neutron clustering: spatial uctuations in multiplying systems at the critical point[END_REF], menant nalement au développement d'un modèle plus rané dans lequel la taille de la population des neutrons est conservée constante, an d'imiter l'eet des méthodes de contrôle de la population [START_REF] De Mulatier | The critical catastrophe revisited[END_REF]. Une caractéristique intéressante de ces modèles est que la plupart des résultats correspondants se généralisent aux problèmes aux valeurs propres résolus par des méthodes Monte-Carlo basées sur les générations, en remplaçant le temps par l'index discret de la génération [START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF] : cela signie aussi que les questions ouvertes pour les modèles en temps continu ont leur contrepartie dans les modèles en générations discrètes. Le principal résultat de ces études est que le `clustering' spatial émerge de la même manière pour les problèmes aux valeurs propres que pour les problèmes cinétiques, à cause des évènements de ssion. En l'absence de contrôle de la population, la variance d'un système critique va diverger, menant possiblement à l'extinction prématurée de la simulation cinétique (catastrophe critique) ; l'application du contrôle de la population empêche l'arrivée de la catastrophe critique. L'impact des précurseurs (en milieu inni et sans contrôle de la population) a été examiné par Houchmandzadeh et al. [START_REF] Houchmandzadeh | Neutron uctuations: The importance of being delayed[END_REF], qui ont montré que l'amplitude des corrélations était largement réduite, et leur évolution temporelle ralentie pour des systèmes proches de la criticité.

Toutes ces investigations sont restreintes à des milieux homogènes et un modèle de transport simplié, ce qui limite leur application à des systèmes plus réalistes, et ce bien que des tendances générales puissent être dégagées. La plus grande de ces limitations vient du fait que ces modèles sont dénis pour des simulations Monte-Carlo analogues, ce qui exclut les méthodes de réduction de variance. Or, la plupart des calculs Monte-Carlo ont couramment recouru aux méthodes non-analogues ; il est donc nécessaire de caractériser précisément l'interaction entre les méthodes de réduction de variance, les méthodes de contrôle de la population et le comportement des corrélations dans les problèmes cinétiques. Enn, bien que dans cette thèse nous nous soyons limités aux simulations cinétiques seules, dans la plupart des applications il est important de prendre en compte l'eet des contre-réactions physiques qui aectent profondément l'évolution des populations de neutrons et de précurseurs. Si le coecient global de contre-réaction est négatif, elles agiront essentiellement comme un mécanisme ramenant le système vers la moyenne, et empêcheront de trop grandes uctuations autour des moyennes. Seules les

  Titre : Analyse des Fluctuations et Corrélations dans les Méthodes Monte-Carlo Cinétiques Mots clés : Monte-Carlo, transport de particules, cinétique, corrélations, uctuations Résumé : Les outils de calcul scientique jouent un rôle déterminant dans la modélisation du comportement des réacteurs nucléaires. La simulation numérique permet de pallier la pénurie de résultats expérimentaux, en particulier en régime non stationnaire (ou cinétique). Traditionnellement, la caractérisation du comportement des c÷urs de réacteurs nucléaires s'appuie sur l'utilisation d'outils de simulation déterministes. Ces codes se fondent sur la résolution numérique de systèmes d'équations discrétisées. Ils permettent d'obtenir rapidement des résultats approchés, dont la validité doit être établie en les comparant à des résultats de référence. La simulation Monte-Carlo, quant à elle, n'introduit quasiment aucune erreur de modélisation et joue ainsi le rôle de méthode de référence. Une simulation Monte-Carlo consiste à échantillonner un grand nombre de trajectoires aléatoires de neutrons dans le réacteur, suivant des lois de probabilité qui dépendent des lois physiques sous-jacentes. La méthode de Monte-Carlo permet un traitement exact, mais lent du système étudié, car l'incertitude statistique sur les résultats est inversement proportionnelle à la racine carrée du nombre d'histoires simulées. L'incertitude peut être réduite en modiant les lois d'échantillonnage, de manière à en préserver le comportement moyen tout en modiant la variance. On parle alors de méthodes de réduction de variance. Jusqu'à très récemment, la simulation Monte-Carlo était appliquée uniquement aux problèmes stationnaires, à cause du coût numérique de son application à la cinétique. En effet, la séparation entre l'échelle de temps des neutrons, dont la durée de vie est très courte, et celle des précurseurs (produits de ssion qui peuvent émettre des neutrons par décroissance), pose un dé considérable. De récentes avancées techniques et algorithmiques permettent cependant d'étendre l'utilisation des simulations Monte-Carlo au domaine des calculs cinétiques. Les méthodes de réduction de variance affectent tous les moments statistiques d'ordre supérieur ou égal à deux. De récents travaux ont mis en avant diérents phénomènes apparaissant dans les simulations cinétiques et trouvant leur origine dans les corrélations spatio-temporelles qui existent entre neutrons appartenant à la même chaîne de ssion. Par exemple, le phénomène du clustering décrit l'existence de collections de neutrons spatialement corrélées formant des amas spontanés à courte distance. L'objectif de ce travail de thèse est de caractériser l'interaction entre les méthodes de réduction de variance et les corrélations spatiotemporelles, et l'eet de cette interaction sur la abilité des incertitudes des simulations Monte-Carlo cinétiques. Pour ce faire, cette thèse est articulée suivant deux axes. Dans une première partie, nous considérons un modèle simplié de réacteur nucléaire, décrit à l'aide de la théorie de la diusion. Nous obtenons une caractérisation analytique de la fonction de corrélations à deux points, décrivant les corrélations entre paires de particules, mais également d'autres observables d'intérêt, telles que la distribution des familles des neutrons dans les chaînes de ssion. Notre étude couvre à la fois le cas de l'évolution de la population neutronique sans contrainte extérieure, et celui de l'évolution sous contrainte, qui représente un modèle simplié de contrôle de la population sur la cinétique des neutrons. Dans un second temps, nous sortons du cadre de la théorie de la diusion. Un nouveau modèle, représentatif du comportement des simulations Monte-Carlo cinétiques telles que réalisées dans le code de référence TRIPOLI-4®, développé au CEA, permet d'évaluer l'impact de diérents algorithmes Monte-Carlo sur les corrélations entre neutrons, dans l'optique de faciliter la détermination d'une stratégie de simulation optimale au sens de la réduction des corrélations, autant dans le cas cinétique que stationnaire. Numerical simulation tools play a crucial role in modelling the behaviour of nuclear reactors. Numerical simulations mitigate the lack of experimental data, in particular in non-stationary (or kinetic) regime. The characterization of the behaviour of nuclear reactors customarily relies on deterministic simulation tools. These codes numerically solve systems of discretized equations.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Power iteration scheme when using ensemble average, with D inactive generations and 1 scoring generations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Schematic illustration of the evolution of neutrons and precursors in the simplest model (anarchic model). At time t = 0, 4 neutrons and 1 precursor are present. Neutrons (red solid lines) diuse and undergo ssion and capture ; precursors (dashed black lines) do not diuse. When they decay, they are replaced by a neutron at the same position. Because of the birth-death process, the total number of neutrons and the total number of precursors uctuate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . 3

 33 Scatter plot of the characteristic time constants of the Green's function (top) and of the one-time, two-point correlation function (bottom), for the anarchic system, up to order k = 5. Top : blue right-pointing triangles represent |ω + k | ; red left-pointing triangles represent |ω - k |. Bottom : blue right-pointing triangles represent |2ω + k | ; black down-pointing triangles represent |ω + k + ω - k | ; red left-pointing triangles represent |2ω - k |. Note that ω - 0 = 0 and 2ω - 0 = 0 cannot be represented in either plot (the system is critical). The dominant mode for each family (k = 0) is the leftmost one. The parameter values are the same as in Fig. 3.2a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Error bars are shown but are barely visible. . . . . . . . . . . . 4.1 Average xation time E[τ p |A](n) (left) and standard deviation σ[τ p |A](n) of the xation time (right) for N = 100 and β = 0.2 in the prompt case. Dashed lines : analytical solution ; circles : Monte Carlo. The Monte Carlo estimates are within 3σ of the analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ratio R = σ[τ p ](n)/E[τ p ](n) of the standard deviation to the average of the xation time, as a function of the family size, in the prompt case, with β = 0.2 and N = 100. . .

  (4.58).. . . . . . . . . . . . . . . . 4.7 One-dimensional slices of the average xation time E[τ |A](n, m) (left) and of the standard deviation of the xation time σ[τ |A](n, m) (right), for N = 100, M = 100, β = 0.2, ν d = 0.5, and λ = 0.1. Solid and dashed lines : numerical results obtained by solving Eqs.(4.76)-(4.77) ; markers : Monte Carlo results are within 3σ of the deterministic solution. 4.8 Pair distance (left) and number of surviving families (right), as calculated by Monte Carlo simulation ; the average xation time is represented by red dashed lines, and the estimated renewal time is represented by blue dashed lines. Parameters are those of Fig. 4.7. . . . .

4. 10

 10 Asymptotic pair correlation function for dierent values of θ, with parameters taken from Table 3.1. Red solid line : analytical solution of the immigration model. Red squares : Monte Carlo simulations of the immigration model. For reference, the results of the Ncontrol scheme are plotted in black triangles. . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Mean-squared pair distance for dierent values of θ, with parameters taken from Table 3.1. Blue dashed line : N M -control model analytical asymptotic value ; blue squares : Monte Carlo result ; red solid line : immigration model analytical solution ; red circles : immigration model Monte Carlo result ; black triangles : N -control model Monte Carlo result. All simulation results are obtained with 10 6 replicas. Error bars are plotted but barely visible. 5.1 Shannon entropy of the ssion source, and the number of distinct families for a critical single-speed box, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and B = 8 3 is also provided. . . . . . . . . . . . . . . 5.2 Shannon entropy of the ssion source, and the number of distinct families for singlespeed transport in a subcritical reactor, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and B = 8 3 is given by the dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Shannon entropy of the ssion source, and the number of distinct families for single-speed transport in a supercritical reactor, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and B = 8 3 is given by the dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  5.1). The values are averaged over 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Space-averaged Feynman moment of the ssion source, as a function of the cell size δ, for N = 10 3 . Dashed black line : square root of ⟨r 2 ⟩ ∞ for branching algorithm. Given that the ⟨r 2 ⟩ ∞ are relatively close to each other for all ve algorithms, we only show the value for the branching algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Convergence of the space averaged Y G,av F with respect to the number of active generations G in terms of the relative bias |Y av F -Y G,av F |/Y av F . The population size is N = 10 3 and B = 10 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Space-averaged ensemble Feynman moments of the ssion source Y av F as a function of the population size N , with bin size δ = 0.5 cm. . . . . . . . . . . . . . . . . . . . . . . 5.12 Equilibrium ensemble normalized variances for the ssion source computed over 10 5 independent replicas. N = 1000 and δ = 0.5 for dierent systems. . . . . . . . . . . . . . 5.13 Scaling of the space-averaged normalized variance g av F as a function of the number of cells L/δ = B, for all 4 systems and algorithms, N = 10 3 , and with 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Equilibrium Feynman moments for the collision counts of neutrons computed for N = 1000 and B = 100 for dierent systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Space-averaged Feynman moments of the collision counts for all ve algorithms, as a function of the cell size δ, for N = 10 3 . Dashed black line : square root of ⟨r 2 ⟩ ∞ for the branching algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  dierent times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Feynman moment of the neutron positions in the supercritical large system with leakage, for dierent times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Entropy of the neutron positions as a function of time in the subcritical congurations with leakage boundary conditions. N 0 = 1000. Blue line : without combing ; orange line : with combing at each time step and on the equilibrium source. . . . . . . . . . . . . . . 6.11 Entropy of the neutron positions as a function of time in the critical congurations with leakage boundary conditions. N 0 = 1000. Blue line : without combing ; orange line : with combing at each time step and on the equilibrium source. . . . . . . . . . . . . . . . . . 6.12 Feynman moment of the equilibrium source in the supercritical case, for N 0 = 1000. Dotted lines gives the Feynman moments before combing, and the solid lines gives the Feynman moments after combing. Red is for neutrons, and blue is for precursors. . . . . 6.13 Example of the Feynman moment of the neutron source when p acc = 10 -3. Dashed red line for Y S,n before combing, solid red line for Y S,n after combing. The dashed black line simply illustrates the deviation from the at spatial shape. . . . . . . . . . . . . . . . . . 6.14 Time evolution of the entropy for neutrons and precursors in the supercritical systems, with N 0 = 10 3 . Dashed lines : large system ; solid lines : small system ; blue for analog decay and red for forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Feynman moment of the neutron population in the supercritical case for N 0 = 1000, for dierent times. We use forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Feynman moment of the precursor population in the supercritical case for N 0 = 1000, for dierent times. We use analog decay. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Feynman moment of the precursor population in the supercritical case for N 0 = 1000, for dierent times. We use forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.18 Time evolution of the entropy for neutrons and precursors in the subcritical systems, with N 0 = 10 3 . Dashed lines : large system ; solid lines : small system ; blue for analog decay and red for forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 Feynman moment of the neutron population in the subcritical case for N 0 = 1000, for dierent times. We use forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20 Feynman moment of the precursor population in the subcritical case for N 0 = 1000, for dierent times. We use forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Feynman moment of the precursor population in the subcritical case for N 0 = 1000, for dierent times. We use analog decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22 Time evolution of the entropy for neutrons and precursors in the critical systems, with N 0 = 10 3 . Dashed lines : large system ; solid lines : small system ; blue for analog decay and red for forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.23 Feynman moment of the precursor population in the critical case for N 0 = 1000, for dierent times. We use forced decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24 Feynman moment of the precursor population in the critical case for N 0 = 1000, for dierent times. We use analog decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.25 Feynman moments of the equilibrium source for system 1, with a total population N = 10 4 and M = 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.26 Feynman moment of the equilibrium source for system 2, with a total population N = 10 4 and M = 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.27 Shannon entropy of the neutron and precursor populations, with a total population N = 10 4 and M = 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.28 Feynman moment of the thermal neutron positions at dierent times, with a total population N = 10 4 and M = 10 4 independent replicas. . . . . . . . . . . . . . . . . . . . .

6. 33

 33 Feynman moments of the precursor population at dierent times, with a total population N = 10 4 and M = 10 4 independent replicas. We consider system 2 for the subcritical transient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.34 Feynman moments of the precursor population at dierent times, with a total population N = 10 4 and M = 10 4 independent replicas. We consider system 2 for the supercritical transient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Photos de la distribution spatiale typique des neutrons dans un calcul aux valeurs propres Monte-Carlo. Gauche : les corrélations sont importantes, et les neutrons sont regroupés. Droite : les corrélations sont faibles, et les neutrons sont répartis uniformément dans l'espace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 . 1

 51 Parameters for the four benchmark congurations. The dominance ratio and the value of the capture cross-sections are taken from Vitali et al. [1]. . . . . . . . . . . . . . . . . . 5.2 Nuclear data of the homogeneous fuel-moderator material. The three energy groups are indexed by g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1 : Snapshots of the spatial positions of neutrons at equilibrium in eigenvalue Monte Carlo simulations. Left : correlations are strong, and neutrons are clustered ; right : correlations are weak, and neutrons are uniformly distributed in space.

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  H. Belanger, T. Bonnet, D. Mancusi, and A. Zoia, The Eects of Branchless Collisions on Neutron Clustering in Proceedings of M&C 2023 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Niagara Falls, Canada, Aug. 13-17th, 2023.

Figure 2 . 1 :

 21 Figure 2.1 : Example of a fission chain as a succession of scattering, capture and fission events, including the precursor decay and the emission of delayed neutrons. The chain is renewed at each fission event, upon which a new branch of the chain is created. Black dots denote neutrons, red dots denote precursors, and connecting lines denote particle flights between collision events.

  being the number of families, leads to the time-dependent Boltzmann equation. The product of n(r, Ω, E, t) with the neutron speed v, which often occurs in the equation, is called angular ux and is denoted by φ(r, Ω, E, t) = vn(r, Ω, E, t).

17

 17

  )

Figure 2 . 2 :Figure 2 . 3 :

 2223 Figure 2.2 : Diagram illustrating the effects of Russian roulette and splitting

Figure 2 . 4 :

 24 Figure 2.4 : Power iteration scheme when using ergodic average, with D inactive generations and G scoring generations

Figure 2 . 5 :

 25 Figure 2.5 : Power iteration scheme when using ensemble average, with D inactive generations and 1 scoring generations.

Figure 3 . 1 :

 31 Figure 3.1 : Schematic illustration of the evolution of neutrons and precursors in the simplest model (anarchic model). At time t = 0, 4 neutrons and 1 precursor are present. Neutrons (red solid lines) diffuse and undergo fission and capture ; precursors (dashed black lines) do not diffuse. When they decay, they are replaced by a neutron at the same position. Because of the birth-death process, the total number of neutrons and the total number of precursors fluctuate.

  (3.44-3.54) by summing the series up to order k = 1000. The resulting correlation function has been carefully veried against Monte Carlo simulations, as shown in Figs. 3.2a and 3.2b. It is more convenient to plot the corrected pair correlation function

  (b) Same as Fig. 3.2a, except that the top graph gives the time dependency for x 1 = 0 and x 2 = -0.02. The bottom graph shows spatial correlations with t 1 = T /2 = 500 and t 2 = 490.

Figure 3 . 2 :

 32 Figure 3.2 : Pair correlation function for a critical population starting with, on average, N = 100 prompt neutrons and M = 10 3 precursors, for a deterministic total population size N = N + M . Parameters are taken as in Table 3.1 for θ = 10 -1 . Blue solid line : analytical results from Eq. (3.35). Black circles : Monte Carlo simulations result with 10 6replicas.

Figure 3 .

 3 Figure 3.2a shows the time shape of the correlation function when particles are detected at the same spatial position x 1 = x 2 (top panel), and the spatial shape when the detection times t 1 = t 2 are the same (bottom panel). The detection positions chosen for Fig. 3.2a are far from the boundaries of the reactor.

Figure 3 .

 3 Figure 3.2b illustrates the time shape of the correlation function for two dierent detection positions (top), and its spatial shape for two dierent detection times (bottom).

Figures

  Figures 3.2a (top panel) and 3.2b (top panel) show that, after an initial transient, t ≲ 50, the time prole of the pair correlation function exhibits a linear build-up. For t 1 > t 2 , correlations saturate. Indeed, the linear build-up describes ssions happening at time t 1 that may contribute to further ssions occurring at a xed t 2 , meaning that obviously ssions occurring at t 1 > t 2 cannot contribute.The analysis of u(x 1 , t 1 , x 2 , t 2 ) shows that the correlation function displays several typical time

  Figures 3.2a (top panel) and 3.2b (top panel) show that, after an initial transient, t ≲ 50, the time prole of the pair correlation function exhibits a linear build-up. For t 1 > t 2 , correlations saturate. Indeed, the linear build-up describes ssions happening at time t 1 that may contribute to further ssions occurring at a xed t 2 , meaning that obviously ssions occurring at t 1 > t 2 cannot contribute.The analysis of u(x 1 , t 1 , x 2 , t 2 ) shows that the correlation function displays several typical time

  (3.35), the twopoint, two-time correlation function involves products of the Green's function, Eq. (3.26), which has the structure of an innite sum of exponential modes with characteristic time constants ω + k and ω - k . The correlation function can then be expressed as an innite sum of exponential modes with characteristic

Figs. 3 .

 3 Figs.[START_REF]Neutronics and thermal-hydraulic coupling in LWR technology[END_REF].2a and 3.2b, the prompt mixing time would be τ D ≃ 41.

Figure 3 . 3 :

 33 Figure 3.3 : Scatter plot of the characteristic time constants of the Green's function (top) and of the one-time, two-point correlation function (bottom), for the anarchic system, up to order k = 5. Top : blue right-pointing triangles represent |ω + k | ; red left-pointing triangles represent |ω - k |. Bottom : blue right-pointing triangles represent |2ω + k | ; black down-pointing triangles represent |ω + k + ω - k | ; red left-pointing triangles represent |2ω - k |. Note that ω - 0 = 0 and 2ω - 0 = 0 cannot be represented in either plot (the system is critical). The dominant mode for each family (k = 0) is the leftmost one. The parameter values are the same as in Fig. 3.2a.

Figure 3 . 4 :

 34 Figure 3.4 : Mean-squared pair distance in a one-dimensional box [-L, L] for critical populations using two sets of parameters taken from Table 3.1. Blue (gray) circles for Monte Carlo simulation with θ = 10 -2 ; blue (gray) solid line for analytical integration of Eq. (3.35) ; black triangles for Monte Carlo simulation with θ = 1 ; black pointed-line for the analytical integration of Eq. (3.10). Error bars are shown but are barely visible.

Figure 3 .

 3 Figure 3.4 shows the time evolution of the mean-squared pair distance for a choice of model parameters. The formula from Eq. (3.56) is compared to the result of the Monte Carlo simulation. Pair distance in Monte Carlo is estimated by taking the ensemble average of the squared distances of all the

First

  , we remark that, for a homogeneous medium with mass preserving boundary conditions, each neutron has an equal probability of fathering the surviving family, implying a xation probability of f (n) = n/N for a family of size n in a total population of N neutrons. To compute the xation time τ p of the population, we rst examine the family-wise conditional xation time. Let A be a family identier ; the other families are ignored. We denote by E[τ p |A](n) the average xation time conditioned to the

Figure 4 . 1 :

 41 Figure 4.1 : Average fixation time E[τ p |A](n) (left) and standard deviation σ[τ p |A](n) of the fixation time (right) for N = 100 and β = 0.2 in the prompt case. Dashed lines : analytical solution ; circles : Monte Carlo. The Monte Carlo estimates are within 3σ of the analytical results

  )where H n is the n-th harmonic number. We nd that, taking N ≫ 1, E[τ p |A](1) ∼ N/β. Comparison with Monte Carlo simulations shows perfect statistical agreement, as shown in Fig.4.1.

Figure 4 . 2 :

 42 Figure 4.2 : Ratio R = σ[τ p ](n)/E[τ p ](n) of the standard deviation to the average of the fixation time, as a function of the family size, in the prompt case, with β = 0.2 and N = 100.

Figure 4 . 3 :

 43 Figure 4.3 : A typical history for a system of 4 neutrons and 1 precursor in the N -control model. Neutrons diffuse (in red line) until they branch or are destroyed by another neutron being produced in the system (these correlated event are shown in vertical blue dashed lines), and precursor lifetime is shown by horizontal black dashed lines.

  First, we denote by N and M the (arbitrary) initial number of neutrons and precursors. The master equation for the total population sizes is easily derived, and because of population control it has a dependency on n only through the initial condition. It reads ∂ ∂t P(n, m, t) = βN ν d,1 P(n, m -1, t) -βN ν d,1 P(n, m, t) + (m + 1)λP(n, m + 1, t) -mλP(n, m, t).

(4. 22 )

 22 We deduce the equation for the precursor population sizes by multiplying Eq. (4.22) by m (resp m) and summing over all possible realizations, which leads to ∂ ∂t n(t) = 0 (4.23a) ∂ ∂t m(t) = βν d,1 N -λm(t).

( 4 .

 4 23b)

3 . 1 . 2 .

 312 In the N -control model, the variance of the neutron population size is zero, as expected, which means in particular that the critical catastrophe is averted. The variance of the precursor population size converges exponentially to m ∞ . It is actually easy to show that if the distribution of the precursor population size is Poissonian at the initial time, then it is Poissonian at all times. First we remind the shape of the Poisson distribution Poiss µ (m) = µ m

( 4 . 23 )

 423 and (4.24), which are closed. On the other hand, Monte Carlo simulations heuristically indicate that, taking M = N/θ and a uniform spatial distribution, the rst moments seem to verify n(x, t) = N 2L (4.38) m(x, t) = M 2L , (4.39)

Figure 4 . 4 :

 44 Figure 4.4 : Spatial dependence of the difference between the pair correlation function evaluated in (x, y = -0.5) and (x = -L = -1, y = -0.5). Black circles : Monte Carlo simulation of the N -control scheme ; blue solid line : analytical solution of the anarchic model. Here, θ = 10 -3 . Parameters are taken from Table 3.1.

Figure 4 . 5 :

 45 Figure 4.5 : A typical history for a system of 2 neutrons and 1 precursor in the N M -control model. Neutrons diffuse (in red solid lines) until they branch or are destroyed by another neutron being produced in the system (these correlated event are shown in vertical blue dashed lines), and precursor lifetime is shown by horizontal black dashed-lines.

  (4.58) is cumbersome. However, we can obtain asymptotic solutions in the form of Fourier series. Let ũ∞ (x, y), v ∞ (x, y) and w∞ (x, y) be the asymptotic shapes of the correlation functions for long times : ũ∞ (x, y) = lim t→∞ ũ(x, y, t) (4.65) v ∞ (x, y) = lim t→∞ v(x, y, t) (4.66) w∞ (x, y) = lim t→∞ w(x, y, t).

( 4 . 67 )

 467 We use the following Fourier decomposition :f (x, y) = +∞ kx,ky=-∞f kx,ky exp (iκ kx (L -x)) exp iκ ky (L -y) .

( 4 . 68 )

 468 Here κ k = kπ/(2L) are the characteristic wave numbers for reection boundary conditions and f (x, y) stands for any of ũ∞ (x, y), v ∞ (x, y) or w∞ (x, y). The coecients satisfy the relation f kx,ky = 1 2π exp (-iκ kx (L -x)) exp -iκ ky (L -y) f (x, y) dx dy.

( 4 .

 4 [START_REF] Hoogenboom | Extensions of the MCNP5 and TRIPOLI4 Monte Carlo Codes for Transient Reactor Analysis[END_REF] by setting the time derivative terms to zero, multiplying by exp(-iκ kx ) exp(-iκ ky y)/(2π), integrating over dx dy and using Eq. (4.69) and Eqs.(4.53). Using the Neumann boundary conditions on f ∞ , the Fourier decomposition then reduces to

(4. 70 )

 70 (a) Asymptotic pair correlation function for θ = 10 -3 and y = 0. (b) Asymptotic pair correlation function for θ = 1 and y = 0.

Figure 4 . 6 :

 46 Figure 4.6 : Asymptotic pair distance for different value of θ. The parameters are taken from Table 3.1. Black triangles : Monte Carlo simulations for the N -control scheme. Blue circles : Monte Carlo simulations of the N M -control scheme, both with 10 6 replicas. Blue solid line : analytical solution of the N M -control scheme Eqs. (4.58).

( 4 .

 4 Figs. 4.6a and 4.6b.

(4. 77 )

 77 Again, Eq. (4.77) is solved numerically by using the boundary condition g 2 (N, M ) = 0. Results are veried against Monte Carlo simulation in Fig. 4.7, with good agreement. Remark also that there is very little dierence between the average xation time for a family initially starting with one precursor or with one neutron, as expected from the shape of f . The standard deviation σ[τ ](n, m) = Var[τ ](n, m) can be shown to behave similarly to the prompt case, in the sense that it has the same order of magnitude as the average xation time : this means that xation can happen early in the simulation, compared to the average xation time.

Figure 4 . 7 :

 47 Figure 4.7 : One-dimensional slices of the average fixation time E[τ |A](n, m) (left) and of the standard deviation of the fixation time σ[τ |A](n, m) (right), for N = 100, M = 100, β = 0.2, ν d = 0.5, and λ = 0.1. Solid and dashed lines : numerical results obtained by solving Eqs.(4.76)-(4.77) ; markers : Monte Carlo results are within 3σ of the deterministic solution.

)

  provided that N is large. Thus, the xation time and the renewal time have a very similar scaling, similarly as in the prompt case. For an illustration of the relation between xation and clustering, Fig. 4.8 displays the pair distance ⟨r 2 (t)⟩ and the number of surviving families ; the xation time and the renewal time are also represented. The time evolution of the number of families and that of the pair distance are of the same order of magnitude. As explained in Sec. 4.1.3, the identication between the xation time and the clustering timescale only holds for unbounded systems.

Figure 4 . 8 :

 48 Figure 4.8 : Pair distance (left) and number of surviving families (right), as calculated by Monte Carlo simulation ; the average fixation time is represented by red dashed lines, and the estimated renewal time is represented by blue dashed lines. Parameters are those of Fig. 4.7.

Figure 4 . 9 :

 49 Figure 4.9 : A typical history for a system with 3 neutrons at t = 0, in the immigration model.

(4. 86 )

 86 (a) Asymptotic pair correlation function for θ = 0.001 and y = 0. (b) Asymptotic pair correlation function for θ = 1 and y = 0.

Figure 4 . 10 :

 410 Figure 4.10 : Asymptotic pair correlation function for different values of θ, with parameters taken from Table 3.1. Red solid line : analytical solution of the immigration model. Red squares : Monte Carlo simulations of the immigration model. For reference, the results of the N -control scheme are plotted in black triangles.

  4.10a) shows that, for systems where θ ≪ 1, the immigration model closely matches the N -control model ; the remarks made about Fig.4.4 also apply to the immigration model.

Figure 4 .

 4 10b provides the same comparison for a larger value of θ. Comparing Fig. 4.10b and Fig. 4.6b, it is clear that the N -control model is better approximated by the N M -control model than by the immigration model. This is unsurprising, in view of the crudeness of the treatment of neutron-precursor correlations in the immigration model.

(4. 88 )

 88 (a) Mean-squared pair distance for θ = 10 -3 . (b) Mean-squared pair distance for θ = 1.

Figure 4 . 11 :

 411 Figure 4.11 : Mean-squared pair distance for different values of θ, with parameters taken from Table 3.1. Blue dashed line : N M -control model analytical asymptotic value ; blue squares : Monte Carlo result ; red solid line : immigration model analytical solution ; red circles : immigration model Monte Carlo result ; black triangles : N -control model Monte Carlo result. All simulation results are obtained with 10 6 replicas. Error bars are plotted but barely visible.

  Several methods have been proposed in order to detect and possibly quench the eects of clustering. Nowak et al. suggested to use Shannon entropy for neutron clustering detection [38], by pointing out that entropy is systematically decreased by the presence of neutron clustering. Inspired by pioneering work in population dynamics by Zhang et al. [81], Sutton introduced the concept of neutron lineage in power iteration and outlined the relation between neutron clustering and xation (i.e. the number of generations at which all neutrons in the Monte Carlo simulation share the same common ancestor) [41, 42]. Equations for the moments of the xation time in kinetic simulations have been derived by us in Ref. 84, which might be used to establish a more rigorous theoretical framework for xation in power iteration. Over the last few years, the issue of spatial correlations has led to a collection of novel, albeit partially contradictory, results. Cosgrove et al. pointed out the impact of clustering, suggesting that it might induce spurious xenon oscillations in depletion calculations coupling Monte Carlo neutronics solvers and Bateman solvers [40]. Fröhlicher et al. suggested that neutron clustering could lead to an apparent bias on the average Monte Carlo tallies, in addition to aecting the estimation of the variance [44]. In contrast, Mickus and Dufek stated that neutron clustering should not be detrimental to criticality Monte

Figure 5 . 1 :

 51 Figure 5.1 : Shannon entropy of the fission source, and the number of distinct families for a critical single-speed box, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and B = 8 3 is also provided.

  The geometrical data are listed in Tab. 5.1. All congurations have leakage boundary conditions at both sides of the slab. The rst two congurations (1 and 2 in Tab. 5.1) are homogeneous, with dierent lengths. The material is a mixture of UO 2 and H 2 O, whose parameters are provided in Tab. 5.2. Congurations 3 and 4 are heterogeneous, with a UO 2 -H 2 O-UO 2 sandwich structure, with lengths given in Tab. 5.1. The respective nuclear data are given in Tab. 5.3 for the fuel and in Tab. 5.4

1 9 . 2 2 9 . 2 3 3 . 2 4 3 .

 9292323 3888852 × 10 -4 1.5003548 × 10 -2 6.9125769 × 10 -3920991 × 10 -4 1.5034394 × 10 -2 7.0670812 × 10 -8396068 × 10 -4 1.1656103 × 10 -2 4.8808556 × 10 -8089301 × 10 -4 1.1361681 × 10 -2 3.4061203 × 10 -2

Figure 5 . 4 :

 54 Figure 5.4 : Shannon entropy as a function of the generations in power iteration, computed for N = 10 3 , B = 10 2 , and for different systems, and averaged over 10 3 independent replicas.

Figure 5 . 5 :

 55 Figure 5.5 : Asymptotic Shannon entropy S ∞ as a function of the bin size δ after discarding an adequate number of generations (cf. 5.1). The values are averaged over 10 4 independent replicas.

Figure 5 .

 5 Figure 5.7 shows the average number of surviving families as a function of generations. At the xation generation, all neutrons in the population are correlated : a larger xation generation should imply weaker correlations. Figures 5.7a and 5.7b contradict this statement : in homogeneous media, branchless collisions without population control lead to the largest xation generation, whereas the highest entropy and pair distance function (hence the weakest spatial correlations) are reached for branchless collisions with combing. Similarly, Fig. 5.7c shows that branching collisions lead to the smallest xation generation,

Figure 5 . 6 :

 56 Figure 5.6 : Square root of pair distance as a function of the generations, computed for N = 10 3 , for different systems, with 10 3 independent replicas.

Figure 5 . 7 :

 57 Figure 5.7 : Average number of surviving families as a function of the generations, computed for N = 10 3 , for different systems, with 10 3 independent replicas.

Figures 5 .

 5 9a and 5.9b show the Feynman moment for the spatially-averaged ssion source dened by

Figure 5 . 8 :

 58 Figure 5.8 : Equilibrium Feynman moment for the fission source computed for N = 1000 and δ = 0.5 for different systems.

Figures 5 .

 5 Figures 5.10a-5.10d show the convergence of the spatially-averaged Feynman moments Y G,av

  average over G generations towards the ensemble spatially-averaged Feynman moments Y av F . For this purpose, we compute the relative bias dened by |Y av F -Y G,av F |/Y av F , as a function of the number of active generations. In all cases, it is clear that the underestimation of errors due to ergodic averages is signicantly smaller when using branchless collisions than when using branching collisions, by about a factor 5 in homogeneous systems and up to a factor 3 in heterogeneous systems. Additionally, the slope of the relative bias as a function of active generations only weakly depends on the details of the sampling algorithm and population control, and is essentially driven by the value of the dominance (a) System 1 with 10 5 realizations. (b) System 2 with 10 5 realizations. (c) System 3 with 10 5 realizations. (d) System 4 with 10 4 realizations.

Figure 5 . 9 :

 59 Figure 5.9 : Space-averaged Feynman moment of the fission source, as a function of the cell size δ, for N = 10 3 . Dashed black line : square root of ⟨r 2 ⟩ ∞ for branching algorithm. Given that the ⟨r 2 ⟩ ∞ are relatively close to each other for all five algorithms, we only show the value for the branching algorithm.

Figure 5 . 10 :

 510 Figure 5.10 : Convergence of the space averaged Y G,avF

Figure 5 .

 5 Figure 5.11 : Space-averaged ensemble Feynman moments of the fission source Y av F as a

  and 5.14d for (a) System 1, with 10 5 independent replicas. (b) System 2, with 10 5 independent replicas. (c) System 3, with 10 5 independent replicas (d) System 4, with 10 4 independent replicas.

Figure 5 . 12 :

 512 Figure 5.12 : Equilibrium ensemble normalized variances for the fission source computed over 10 5 independent replicas. N = 1000 and δ = 0.5 for different systems.

Figure 5 . 13 :

 513 Figure 5.13 : Scaling of the space-averaged normalized variance g av F as a function of the number of cells L/δ = B, for all 4 systems and algorithms, N = 10 3 , and with 10 4 independent replicas.

(a) System 1 with 10 5

 5 independent replicas. (b) System 2 with 10 5 independent replicas. (c) System 3 with 10 4 independent replicas. (d) System 4 with 10 4 independent replicas.

Figure 5 . 14 :

 514 Figure 5.14 : Equilibrium Feynman moments for the collision counts of neutrons computed for N = 1000 and B = 100 for different systems.

(a) System 2 ,

 2 with 10 4 replicas. (b) System 4, with 10 3 replicas.

Figure 5 . 15 :

 515 Figure 5.15 : Space-averaged Feynman moments of the collision counts for all five algorithms, as a function of the cell size δ, for N = 10 3 . Dashed black line : square root of ⟨r 2 ⟩ ∞ for the branching algorithm.

Figure 5 . 16 :

 516 Figure 5.16 : Equilibrium normalized variance for the collision counts of neutrons, computed for N = 1000, δ = 0.5, and with 10 4 independent replicas, for the two heterogeneous systems.

(a) System 1 ,

 1 with 10 5 independent replicas. (b) System 2, with 10 5 independent replicas. (c) System 3, with 10 5 independent replicas. (d) System 4, with 10 4 independent replicas.

Figure 5 . 17 :

 517 Figure 5.17 : Space-averaged equilibrium normalized variance for the collision counts of neutrons as a function of bin size, computed for N = 1000 and B = 100 for different systems.

Figure 5 . 18 :

 518 Figure 5.18 : Shannon entropy of the fission source, and the number of distinct families for a critical continuous energy box, for both the branching and branchless simulation methods. The ideal value of the entropy for N = 10 3 and = 8 3 is also provided.

)

  where r i is the center of the i-th spatial cell, E[•] denotes the ensemble average over M independent replicas of the time-dependent observable O, and V[O] is the ensemble unbiased estimator of the variance of O. The interpretation of the Feynman moment presented in Chapter 5 still holds.

( a )Figure 6 . 1 :

 a61 Figure 6.1 : Shannon entropy as a function of time. The solid lines represent Monte Carlo results, and dashed black lines the ideal Shannon entropy computed for different population sizes.

( a )Figure 6 . 2 :

 a62 Figure 6.2 : Feynman moment for the neutron positions in the supercritical one-speed configuration, with initial population size N 0 = 250, at different times.

( a )Figure 6 . 4 :Figure 6 . 5 :

 a6465 Figure 6.4 : Shannon entropy as a function of time when applying weight combing at each time step. The solid lines represent Monte Carlo results, and the pointed lines the ideal Shannon entropy computed for the total neutron population at a given time.

( a )Figure 6 . 6 :Figure 6 . 7 :

 a6667 Figure 6.6 : Feynman moment for the equilibrium source in the supercritical one-speed configuration, with L = 400, with initial population size N 0 = 250, at different times.

( a )

 a Feynman moments of the neutron positions when not applying combing. (b) Feynman moments of the neutron positions when applying combing.

Figure 6 . 8 :

 68 Figure 6.8 : Feynman moment of the neutron positions in the supercritical small system with leakage, for different times.

  Finally, the value of the Feynman moment again overshoots the value of the average neutron weight, indicating non-Poisson uctuations of the spatial distribution.

( a )Figure 6 . 9 :

 a69 Figure 6.9 : Feynman moment of the neutron positions in the supercritical large system with leakage, for different times

Figure 6 . 10 :

 610 Figure 6.10 : Entropy of the neutron positions as a function of time in the subcritical configurations with leakage boundary conditions. N 0 = 1000. Blue line : without combing ; orange line : with combing at each time step and on the equilibrium source.

  ± 5 pcm 0.21292 ± 5 pcm L = 100 (crit) 2.54103 1.00007 ± 7 pcm 0.21286 ± 5 pcm L = 400 (sup)2.1 1.03111 ± 2 pcm 0.01800 ± 2 pcm L = 400 (sub)1.9 0.93295 ± 2 pcm 0.01796 ± 2 pcm L = 400 (crit) 2.03655 0.99993 ± 5 pcm 0.01802 ± 5 pcm Table6.2 : Fission yields for the configurations with delayed neutrons, and corresponding multiplication factors and leakage probability.

Figure 6 . 12 :

 612 Figure 6.12 : Feynman moment of the equilibrium source in the supercritical case, for N 0 = 1000. Dotted lines gives the Feynman moments before combing, and the solid lines gives the Feynman moments after combing. Red is for neutrons, and blue is for precursors.

Figure 6 . 13 :Figure 6 . 14 :

 613614 Figure 6.13 : Example of the Feynman moment of the neutron source when p acc = 10 -3. Dashed red line for Y S,n before combing, solid red line for Y S,n after combing. The dashed black line simply illustrates the deviation from the flat spatial shape.

Figure 6 . 18 :Figure 6 . 19 :

 618619 Figure 6.18 : Time evolution of the entropy for neutrons and precursors in the subcritical systems, with N 0 = 10 3 . Dashed lines : large system ; solid lines : small system ; blue for analog decay and red for forced decay.

Figure 6 . 22 :Figure 6 . 23 :

 622623 Figure 6.22 : Time evolution of the entropy for neutrons and precursors in the critical systems, with N 0 = 10 3 . Dashed lines : large system ; solid lines : small system ; blue for analog decay and red for forced decay.

( a )

 a Entropy of the total neutron population. (b) Entropy of the total precursor population.

Figure 6 . 27 :

 627 Figure 6.27 : Shannon entropy of the neutron and precursor populations, with a total population N = 10 4 and M = 10 4 independent replicas.

Figure 6 . 28 :

 628 Figure 6.28 : Feynman moment of the thermal neutron positions at different times, with a total population N = 10 4 and M = 10 4 independent replicas.

Figure 6 . 29 :

 629 Figure 6.29 : Feynman moments of the collision density at different times, with a total population N = 10 4 and M = 10 4 independent replicas. Squares are for the fast group, circles for the epithermal group, and triangles for the thermal group. Blue is for t = 0.001 s ; red is for t = 0.1 s ; orange is for t = 1 s, and green for t = 10 s.

Figure 6 . 31 :

 631 Figure 6.31 : Shannon entropy of the neutron and precursor populations when changing reactivity at t = 0, with a total population N = 10 4 and M = 10 4 independent replicas. Blue for system 1 and red for system 2 ; solid lines for the reactivity removal, and cross markers for the reactivity insertion.

Figure 6 . 32 :

 632 Figure 6.32 : Feynman moments of the collision density at different times, with a total population N = 10 4 and M = 104 independent replicas. We consider system 2 for the supercritical and for the subcritical transients. Squares are for the fast group, circles for the epithermal group, and triangles for the thermal group. Blue is for t = 0.001 s ; red is for t = 0.1 s ; orange is for t = 1 s, and green for t = 10 s.

  (a) Feynman moments of the precursor population at t = 1 s. (b) Feynman moments of the precursor population at t = 10 s.

Figure 6 . 33 :

 633 Figure 6.33 : Feynman moments of the precursor population at different times, with a total population N = 10 4 and M = 10 4 independent replicas. We consider system 2 for the subcritical transient.

  (a) Feynman moments of the precursor population at t = 1 s. (b) Feynman moments of the precursor population at t = 10 s.

Figure 6 . 34 :

 634 Figure 6.34 : Feynman moments of the precursor population at different times, with a total population N = 10 4 and M = 10 4 independent replicas. We consider system 2 for the supercritical transient.

  The model introduced in Chapter 3 has been rened in Chapter 4 by taking into account several idealized sampling strategies mimicking the eects of population control. These schemes break the statistical independence of the neutron ssion chains, which forced us to abandon the Pál and Bell backward formalism in favour of the forward formalism. After discussing the derivation of the master equation for the simplied, yet instructive, case where precursors are neglected, we have developed three dierent population-control schemes : the N -control model, where the neutron population is kept constant, but no constraint is applied to the precursor population ; the N M -control model, both the neutron and precursor populations are controlled ; and the (approximate) immigration model, where precursors are described as a uniform Poisson neutron source whose intensity is equal to the asymptotic precursor decay rate at equilibrium. We have shown that the master equation related to the N -control model leads to an innite hierarchy of moment equations ; nonetheless, the behavior of the average particle number and the pair correlation function can be determined using Monte Carlo simulation.For the N M -control model, the hierarchy of moment equation is closed : we have derived asymptotic solutions for the average particle number and the pair correlation function, and we have successfully compared them against Monte Carlo simulations. Finally, we have shown that the third model, although approximate, is in fairly good agreement with the N -control and N M -control model under reasonable assumptions on the values of the physical constants. In all cases, we showed that the introduction of population control prevents the occurrence of the critical catastrophe, and thus quenches clustering.Within the same theoretical framework, we have also gained insight on the behaviour of kinetic Monte Carlo simulations by investigating family xation, i.e. the eventual survival of a single particle family, which provides a complementary point of view with respect to the pair correlation function. In particular, we have addressed the distribution of the xation time, i.e. the time at which all the particles belong to the same family, and its relation to the typical time-scales leading to spatial clustering. We have shown that these quantities, although intimately related, have subtle dierences : the xation time is by construction insensitive to nite-size eects, whereas the evolution of clustering is deeply aected by the presence of boundaries.

A. 1 .

 1 IntroductionDans le contexte de la sureté des réacteurs nucléaires, le développement d'outils de simulations rapides et ecaces pour le couplage entre les solveurs pour le ux neutronique et les codes de thermohydraulique et de thermo-mécanique, dans des conditions stationnaires et transitoires, est le sujet de vastes programmes de recherche dans plusieurs institutions majeures 1 .

  for Evaluation And Validation of Reactor Simulations) 3 , et Watts Bar (TVA Watts Bar Unit 1 Multi-Physics Benchmark)

  ainsi obtenus : les codes Monte-Carlo et les codes déterministes se complètent les uns les autres et sont également nécessaires. Principalement en raison du temps de calcul nécessaire et des limitations en puissance de calcul, en physique des réacteurs les codes Monte-Carlo ont longtemps été cantonnés exclusivement à l'étude des problèmes stationnaires à température xée. Cependant, les avancées récentes tant en termes d'algorithmes qu'en termes de puissance de calcul ont permis de se diriger vers une nouvelle classe d'applications pour les simulations Monte-Carlo, et plus spéciquement vers leur application aux problèmes multi-physiques. D'énormes progrès ont par exemple été eectués dans le domaine de la déplétion du combustible, lors de laquelle le comportement de la densité de neutrons est couplé avec la transmutation du matériau irradié, dont l'évolution est dictée par les équations généralisées de Bateman ; des codes Monte-Carlo ont été ecacement couplés avec des solveurs déterministes pour les équations de Bateman, et la stabilité des schémas de simulation correspondants a été largement étudiée [1116].Ces dernières années ont vu l'émergence d'un intérêt grandissant pour une nouvelle classe de méthodes Monte-Carlo dévolues aux simulations cinétiques (i.e dépendant du temps), dont le temps de calcul était bien trop élevé jusqu'à il y a peu. Encouragées par les travaux pionniers de Sjenitzer et Hoogenboom[START_REF] Sjenitzer | A Monte Carlo method for calculation of the dynamic behaviour of nuclear reactors[END_REF][START_REF]Dynamic Monte Carlo Method for Nuclear Reactor Kinetics Calculations[END_REF], d'ingénieuses méthodes de réduction de variance et de contrôle de la population ont vu le jour, destinées à traiter ecacement les problèmes dépendant du temps. D'abord testées dans des modèles simpliés, ces méthodes ont ensuite été introduites avec succès dans la simulation de réacteurs nucléaires réalistes prenant en compte l'eet des contre-réactions thermo-hydrauliques[1926].A.1.2 . Corrélations et uctuations dans les simulations Monte-Carlo cinétiquesÉtant donné l'apparition relativement récente de ces nouvelles méthodes, un cadre complet pour l'étude des simulations Monte-Carlo dépendantes du temps manque encore, malgré de prometteuses investigations préliminaires. Comme les simulations Monte-Carlo doivent servir de références auxquelles d'autres méthodes simpliées seront comparées, la précision et l'exactitude des simulations Monte-Carlo sont cruciales. En particulier, la abilité des résultats dépend directement de la abilité des intervalles de conance associés, qui sont typiquement estimés en utilisant des techniques d'inférence statistique (en estimant la variance des échantillons Monte-Carlo). La variance d'échantillon capture les uctuations dans les scores dues à la nature stochastique des histoires. Si ces histoires sont indépendantes et si les uctuations ne sont pas pathologiques, la quantication de la variance de l'échantillon est directe.Cependant, les simulations cinétiques posent des dés bien particuliers.La condition initiale pour la densité de neutrons et de précurseurs dans les problèmes dépendant du

Figure A. 1 :

 1 Figure A.1 : Photos de la distribution spatiale typique des neutrons dans un calcul aux valeurs propres Monte-Carlo. Gauche : les corrélations sont importantes, et les neutrons sont regroupés. Droite : les corrélations sont faibles, et les neutrons sont répartis uniformément dans l'espace.

  et al. ont calculé les coecients d'auto-corrélation entre les cellules spatiales an d'analyser le comportement des corrélations générationnelles [3537].

A. 1 . 3 .•

 13 But et structure de la thèseInspirés par ces considérations, dans cette thèse nous avons développé un cadre cohérent et unié pour l'analyse des uctuations et des corrélations dans les simulations Monte-Carlo cinétiques, dans le but d'assurer la abilité des estimations de l'incertitude statistique dans cette classe de simulations. En particulier, nous nous sommes concentrés sur l'interaction entre le comportement des uctuations et des corrélations, et l'application de méthodes de réduction de variance et de contrôle de la population, ainsi qu'avec les stratégies d'échantillonnage spéciquement dénies pour les problèmes cinétiques et stationnaires (ou aux valeurs propres), ces derniers étant essentiels pour l'échantillonnage de la source d'équilibre des neutrons et des précurseurs.Ce travail est structuré de la manière suivante :• Tout d'abord, dans les Chapitres 1 et 2, nous avons présenté l'état de l'art des simulations Monte-Carlo pour le transport des neutrons, et nous avons porté une attention particulière aux méthodes utilisées pour les simulations cinétiques.• Dans la Partie I (Chapitres 3 et 4), nous avons présenté nos résultats dans le cadre d'un modèle simplié de simulation Monte-Carlo analogue d'un processus de diusion, dans lequel une population de neutrons et de précurseurs évolue en temps en suivant un mouvement Brownien branchant.Nous avons caractérisé les corrélations induites par la ssion dans ce système, et nous avons dérivé des grandeurs caractéristiques an de faciliter l'analyse de simulations plus réalistes. Nous avons aussi étudié l'eet d'un mécanisme idéalisé de contrôle de la population dans le cas de simulations analogues. Dans la Partie II (Chapitres 5 et 6), nous avons présenté nos résultats concernant les simulations Monte-Carlo cinétiques non-analogues, en utilisant un large panel de techniques de réduction de variance et de contrôle de la population. Nous avons proposé des estimateurs adéquats pour l'analyse des corrélations dans ces systèmes, et discuté de l'eet des méthodes de réduction de variance et de contrôle de la population sur les uctuations et les corrélations.Dans la première partie de la thèse (Chapitres 3 et 4), nous avons examiné le comportement des uctuations et des corrélations dans les simulations Monte-Carlo cinétiques dans le cas simplié où l'échantillonnage est analogue (ce qui signie que nous n'utilisions pas de réduction de variance) et en utilisant l'approximation de la diusion. Ces choix ont été eectués de manière à proposer un cadre permettant de calculer des solutions analytiques exactes pour les équations des moments décrivant la fonction de corrélation à deux points, et de les comparer à des simulations Monte-Carlo.Dans ce but, dans le Chapitre 3 nous avons introduit un modèle stochastique dans lequel une collection de particules suit un mouvement Brownien couplé à un processus de vie et de mort de Galton-Watson imitant les évènements de collisions. Lors de ces évènements, les particules peuvent être capturées (et disparaissent), ou induire une ssion (et donnent naissance à un nombre aléatoire de descendants). Le système est unidimensionnel avec des conditions de bord de réexion préservant la masse. Les précurseurs sont modélisés en utilisant un retard exponentiellement distribué entre l'évènement de ssion dont ils sont issus et l'émission de neutrons supplémentaires depuis le point de collision. Malgré ces approximations extrêmement fortes, ce modèle préserve les éléments clés du transport dépendant du temps dans un milieu multiplicatif, et ore ainsi d'importantes informations sur le comportement des uctuations et des corrélations dans les simulations Monte-Carlo analogues. Dans ce contexte, nous avons utilisé le formalisme rétrograde de Pál-Bell pour dériver les équations des moments, en particulier pour la densité moyenne de particules et la fonction de corrélation à deux points. Nous avons résolu analytiquement ces équations, et nous avons comparé ces résultats à des simulations Monte-Carlo analogues, utilisées comme référence. Une attention particulière a été portée au cas critique, dans lequel les naissances par ssion sont compensées (en moyenne) par les morts dues à la capture, ce qui est crucial pour le contrôle des réacteurs. Nous avons identié les échelles de temps typiques selon lesquelles évoluent les corrélations : le temps pour que les corrélations issues de la dynamique prompte se stabilisent, le temps nécessaire pour que les corrélations atteignent leur forme spatiale asymptotique, et le temps nécessaire pour observer des uctuations catastrophiques de la taille de la population. Le modèle introduit dans le Chapitre 3 a été rané dans le Chapitre 4 en prenant en compte des stratégies idéalisées de contrôle de la population imitant l'eet du contrôle de la population dans les simulations non-analogues. Ces schémas brisent l'indépendance statistique entre diérentes chaînes de ssion, forçant l'abandon du formalisme de Pál-Bell au prot du formalisme direct. Après avoir présenté la dérivation de l'équation maîtresse pour le cas simplié dans lequel les précurseurs sont négligés, nous avons développé trois modèles diérents de contrôle de la population : le modèle avec contrôle de N , dans lequel la population de neutrons est maintenue constante tandis que les précurseurs évoluent librement ; le contrôle de N et M , dans lequel une contrainte est appliquée séparément sur les neutrons et les précurseurs ; et un modèle d'immigration dans lequel les précurseurs sont modélisés comme une source Poissonnienne uniforme dont l'intensité est égale au taux asymptotique de décroissance des précurseurs à l'équilibre. Nous avons montré que l'équation maîtresse du modèle de contrôle de N menait à une hiérarchie innie d'équations des moments ; néanmoins, le comportement du nombre moyen de particules et de la fonction de corrélation de paires peut toujours être déterminé en utilisant les simulations Monte-Carlo. Pour le modèle de contrôle de N et M , la hiérarchie d'équations des moments est fermée : nous avons déterminé des solutions asymptotiques pour la fonction de corrélation de paires, qui ont été comparées avec succès aux simulations Monte-Carlo. Finalement, nous avons montré que le troisième modèle, quoiqu'approché, est en relativement bon accord avec les deux autres modèles pour certaines valeurs (raisonnables) des paramètres physiques. Dans tous les cas, nous avons montré que l'introduction du contrôle de la population empêche la catastrophe critique, et donc limite l'apparition du `clustering'. Dans le même cadre théorique, nous avons aussi caractérisé le comportement des simulations Monte-Carlo en étudiant le processus de xation, c'est-à-dire la survivance d'une seule famille de particules à un temps donné, orant ainsi un point de vue complémentaire à la fonction de corrélation de paires. En particulier, nous avons étudié la distribution du temps de xation, c'est-à-dire le temps auquel toutes les particules appartiennent à la même famille, et ses liens avec les grandeurs caractéristiques du `clustering'. Nous avons montré que ces quantités, quoiqu'intimement connectées, ont de subtiles diérences : le contrôle de population utilisées, et la nature des conditions aux bords. Dans certains cas, les oscillations de l'entropie pouvaient perdurer sur de longues échelles de temps. Enn, nous avons étudié les corrélations spatio-temporelles apparaissant dans les cas tests homogènes plus réalistes introduits dans le Chapitre 5. Nous avons détaillé comment les corrélations spatiales se développent avec le temps. Nos résultats suggèrent que l'eet des insertions de réactivité sur les corrélations spatiales est marginal, au moins dans le régime des petites perturbations. De plus, la catastrophe critique est eectivement évitée grâce à l'usage de méthodes non-analogues. Le travail eectué lors de cette thèse a permis de clarier le comportement des uctuations et des corrélations dans les simulations Monte-Carlo dépendantes du temps, en commençant par des modèles très simpliés que nous avons progressivement rendu plus complexes an d'agrandir le domaine de validité de nos résultats. An d'obtenir une compréhension plus précise de l'interaction entre les méthodes d'échantillonnage et les corrélations, nous avons conservé nos cas tests bien plus simples que pour des applications réalistes typiquement traitées par des codes Monte-Carlo industriels. D'un côté, cela nous a permis d'estimer et d'analyser des observables qui auraient autrement été hors de notre portée, en raison du coût de calcul, telles que la fonction de corrélations à deux points. De l'autre côté, nous sommes conscients que les conclusions obtenues à partir de congurations uni-dimensionnelles en utilisant des sections multi-groupes peut ne pas être représentatif du comportement observé dans des systèmes tridimensionnels utilisant des données nucléaires en énergie continue. De plus, il est important de noter que les méthodes Monte-Carlo cinétiques sont encore relativement jeunes et incomprises : de nouvelles méthodes de réduction de variance ou de contrôle de la population pourraient apparaître, soit spéciquement conçues pour les méthodes dépendantes du temps, soit adaptées des simulations stationnaires. De tels changements pourraient nous obliger à réviser les conclusions présentées ici, particulièrement pour le cas des simulations cinétiques discuté dans le Chapitre 6. Plusieurs chemins sont naturellement ouverts par les résultats de cette thèse. Comme brièvement noté dans le Chapitre 5, remplacer les sections multi-groupes par des données nucléaires en énergie continue pourrait avoir un eet signicatif sur le comportement des méthodes de réduction de variance et les corrélations et uctuations induites par celles-ci. Une étude plus vaste en trois dimensions et en énergie continue est donc nécessaire, ainsi que le développement d'outils de diagnostic adaptés à la détection des corrélations, an de mieux contrôler l'augmentation du coût en temps de calcul et en mémoire. Une autre possibilité concerne l'analyse de l'interaction entre l'échelle spatiale des corrélations et l'échelle spatiale des observables Monte-Carlo. En se basant sur les résultats présentés dans les Chapitres 5 et 6, il apparaît que l'émergence des corrélations spatiales est intimement reliée à la compétition entre la taille linéaire des regroupements de particules et celle des régions dans lesquelles les observables sont estimées. Une interaction similaire pourrait aussi exister pour les échelles temporelles : celles des corrélations et celles des observables dans les simulations cinétiques ; cette conjecture devrait être étudiée numériquement. Il serait aussi intéressant de développer un cadre théorique permettant d'adapter la taille des cellules (spatiales ou temporelles) pour les observables Monte-Carlo an de réduire l'eet des corrélations. En rapport avec les méthodes de réduction de variance, il serait intéressant d'eectuer une étude systématique des schémas à variance zéro telle que celle entamée dans la Ref. 100, an d'obtenir un cadre cohérent pour une stratégie CADIS (Consistent Adjoint-Driven Importance Sampling). Ces méthodes pourraient avoir un immense potentiel dans le cadre des simulations Monte-Carlo cinétiques, de la même manière que pour les problèmes stationnaires de radioprotection, dans lesquels les méthodes CADIS sont maintenant couramment utilisée.

Table des gures 1

 des1 .1 Snapshots of the spatial positions of neutrons at equilibrium in eigenvalue Monte Carlo simulations. Left : correlations are strong, and neutrons are clustered ; right : correlations are weak, and neutrons are uniformly distributed in space. . . . . . . . . . . . . . . . . .2.1Example of a ssion chain as a succession of scattering, capture and ssion events, including the precursor decay and the emission of delayed neutrons. The chain is renewed at each ssion event, upon which a new branch of the chain is created. Black dots denote

neutrons, red dots denote precursors, and connecting lines denote particle ights between collision events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Bonnet, D. Mancusi, and A. Zoia, "Space and time correlations for diusion models with prompt and delayed birth-and-death events", Phys. Rev. E, vol. 105, no. 6, p. 064105, Jun. 2022.[Online]. T. Bonnet, D. Mancusi, and A. Zoia, The Statistics of Family Histories in Kinetic Monte Carlo Simulations, in Proceedings of M&C 2023 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Niagara Falls, Canada, Aug.

	Available : https://link.aps.org/doi/10.1103/PhysRevE.105.064105
	13-17th, 2023.

•

  The collided nuclide is selected by drawing ξ ∼ U[0, 1[ and comparing ξ to the probability p i (E) dened in Eq. (2.2). Once the nuclide is sampled, the reaction channel k is determined by drawing again ξ ∼ U[0, 1[ and comparing ξ to the probability p i,k (E) dened by

  being able to sample from Eqs.(2.15) and (2.16) at equilibrium. As such, it is necessary to determine the dominant eigenpair {k 0 , φ k 0 } of Eq.

	Algorithm 1: Kinetic Monte Carlo scheme
	1 Choose tentative source Q;
	2 for each replica m do
	3	Inject Q into power iteration calculation;
	4	Iterate until convergence to the stationary state;
	5	Sample the kinetic source S m ;
	6	Normalize S m to desired total weight;
	7	for each time-step do
	8	Transport particles;
	9	Apply population control;
	10	Save surviving particles for next time-step;
	11	Accumulate statistics;
	be detailed in the following.
		2.4 . Determining the fundamental eigenpair : the power iteration method
		Often, the source for time-dependent problems corresponds to the reactor being in a critical state,
	which amounts to

  3.2.6 . Exact solutions for the anarchic model with a critical sourceFor the critical anarchic model with a critical source, the solution of the pair correlation function in Eq.(3.35) can be derived explicitly. For t 1 < t 2 , the pair correlation function u(x 1 , t 1 , x 2 , t 2 ) can be decomposed as follows :

  (4.31) describes the evolution of the discrete occupation probability. Correspondingly, we can derive the average number of neutrons in V i at time t from

.2) Equation (4.2) is hardly amenable to analytical solutions, but insights about the statistical behaviour of the neutron population can nonetheless be gained by investigating the statistical moments of the neutron population. 4.1.2 . Derivation of the spatial moments Equation E[n i ](t) = n n i P(n, t).

  The eigenvalues {r 1 , r 2 , r 3 }

	read
	r 1 ≃ -2βν d,1	(4.61)
	r 2 ≃ -βν d,1

  t),(4.84) where we recognize τ n from Eq. (4.59a). Thus, in this model, neutron pair correlations relax with a time constant governed by τ n .

	Equations (4.82) and (4.84) have the same form as Eqs. (4.53a) and (4.58a), provided that one
	replaces Q I (x) by its expression, Eq. (4.79). These equations can be solved analytically. In particular,
	taking a uniform initial condition for n(x, t), we obtain
	n(x, t) =	N 2L	(4.85)

Table 5 .

 5 1 : Parameters for the four benchmark configurations. The dominance ratio and the value of the capture cross-sections are taken from Vitali et al.[START_REF] Vitali | Eigenvalue separation and eigenmode analysis by matrix-lling Monte Carlo methods[END_REF].

	Number	Type	Inactive generations	Lengths Dominance (cm) ratio	k eff
	1	Homogeneous	100	50	0.9832	1.00003 ± 1 pcm
	2	Homogeneous	200	100	0.9957	0.99986 ± 1 pcm
	3	Heterogeneous	20	20/2/20	0.7908	0.99999 ± 3 pcm
	4	Heterogeneous	1000	20/20/20	0.9971	0.99987 ± 5 pcm

Table 6 .

 6 However, contrary to the supercritical case, the Monte Carlo entropy coincides with the ideal entropy, suggesting that, in this case, 1 : Parameters for the one-speed problem with vacuum boundary conditions combing does not introduce spatial correlations. This conclusion is further substantiated by remarking

	System	ν p	k eff	Leakage probability τ th [s]
	L = 100 (sup)	3.2	1.25925 ± 8 pcm	0.21291 ± 5 pcm	0.2900
	L = 100 (sub)	2.2	0.86576 ± 5 pcm	0.21294 ± 5 pcm	1.7341
	L = 100 (crit) 2.54103 1.00007 ± 7 pcm	0.21286 ± 5 pcm	0.6418
	L = 400 (sup)	2.5	1.22759 ± 13 pcm	0.01795 ± 5 pcm	0.6944
	L = 400 (sub)	1.75	0.85932 ± 7 pcm	0.01797 ± 5 pcm	0.4458
	L = 400 (crit) 2.03655 0.99993 ± 5 pcm	0.01802 ± 5 pcm	9.4834

  pcm 1.08875 × 10 -3 -1.1219 × 10 -3 2 0.99986 ± 1 pcm 1.13902 × 10 -3 -9.61192 × 10 -4

	System number	k eff	δρ (super)	δρ (sub)
	1	1.00003 ± 1

Table 6 .

 6 6 : Amplitude of the reactivity insertions for each case. family 1 family 2 family 3 family 4 family 5 family 6

	w eq j	0.2373	0.4991	0.1292	0.1155	0.0169	0.0021

Table 6 .

 6 7 : Expected fractional weight in the precursor source. The families are indexed by j.

The (classical) phase space includes position (three variables) and velocity (three variables). Velocity can be equivalently expressed using direction (two variables) and energy (one variable).https://world-nuclear-news.org/Articles/Federal-funding-for-reactor-modelling-initiative. http://crpg.mit.edu/sites/default/files/css_injector_images_image/BEAVRS_2.0.2_spec.pdf. https://www.oecd-nea.org/jcms/pl_32202/tva-watts-bar-unit-1-multi-physics-benchmark.

There is no formal restriction preventing from using independent replicas in the context of power iteration. The main reason that led Monte Carlo practitioners to rely on ergodic averaging stems from considerations of computational cost : independent replicas would involve a total cost much higher than ergodic averaging, and this was prohibitive in the early history of Monte Carlo methods.

In these simulations, precursors were neglected.

With the only possible exception of start-up or shut-down regimes, where the reactor is operated at very low power.

We assume for the sake of simplicity a perfect detection process : in our model, all the neutrons falling within a given region V i , i ∈ {1, 2}, are counted.

Observe that branchless collisions reduce to branching collisions with implicit capture in regions without fission.

To partially overcome this limitation, Nowak et al. have proposed to use the spatial moments of the Shannon entropy, weighted by Legendre polynomials along the Cartesian axes[START_REF] Nowak | Monte Carlo power iteration: Entropy and spatial correlations[END_REF] : using spatial symmetries occurring in the simulated system, deviations of the moments of the Shannon entropy may help in revealing the emergence of clustering.

Given that reaction rates are akin to partial collision counts, one could also analyze their Feynman moments, which will not attempted here, for the sake of conciseness.

The derivation of Eq. (5.19) by Nowak et al. assumed analog transport ; its use for predicting the fission source entropy in a weighted game is perhaps inappropriate, but we use it here nonetheless, as an approximation.

The multiplication factor k eff of the critical cases is as close as possible to 1, within 1σ uncertainties.

Voir par exemple les initiatives aux Etats-Unis, comme le projet CESAR (cesar.mcs.anl.gov),le consortium CASL (www.casl.gov), ou encore les projets européens FP7, HPMC, et McSAFE (cordis.europa.eu/ projects).

2 https://world-nuclear-news.org/Articles/Federal-funding-for-reactor-modelling-

initiative.[START_REF]Neutronics and thermal-hydraulic coupling in LWR technology[END_REF] http://crpg.mit.edu/sites/default/files/css_injector_images_image/BEAVRS_2.

0.2_spec.pdf.[START_REF] Dulla | The quasi-static method revisited[END_REF] https://www.oecd-nea.org/jcms/pl_32202/tva-watts-bar-unit-1-multi-physics-benchmark.

Il n'y a pas de restrictions de principe empêchant d'utiliser des répliques indépendantes dans l'itération de la puissance. La principale raison qui a mené les pratiquants du Monte-Carlo à adopter la moyenne ergodique vient du temps de calcul : utiliser des répliques indépendantes impliquerait un temps de calcul largement plus élevé qu'en utilisant la moyenne ergodique, ce qui était décourageant dans les premiers temps du Monte-Carlo.

Acknowledgements

This framework sets the bases to fully characterize the stochastic evolution of a collection of neutron and precursors when no external constraints are imposed. In the next chapter, we will focus on the space and time correlations in the presence of idealized population control algorithms. Single-speed transport in the critical cube leads to a branchless weight modier of 1 ; thus, the weight of particles will never change, so no splitting or roulette will ever occur. This fact, combined with the reective boundary conditions, means that no particles will ever be removed from the system, so the number of independent particle families will remain constant instead of decreasing as it would be the case otherwise. The decrease in the number of particle families has been shown to be one of the main causes of clustering in Monte Carlo simulations [START_REF] Sutton | Toward a More Realistic Analysis of Neutron Clustering[END_REF], and the fact that the number of families is constant, when using branchless collisions, explains the absence of clustering. Thus, single-speed transport in a critical reactor is a somewhat pathological case that does not need any population control mechanism when using branchless collisions. We chose to also examine a subcritical and supercritical version of the reactor, where the ssion yield was used to control k e . The respective parameters were ν = 2, k e = 0.8 and ν = 3, k e = 1.2. For these cases, we considered the use of combing, splitting, and sampling without replacement (WOR) with splitting. The entropy and number of families are presented in Fig. 5.2 for the subcritical case, and Fig. 5.3 for the supercritical case. In both the subcritical and supercritical congurations, all branchless collision variants yield larger numbers of families, larger values of the entropy and therefore seem to be much more eective at reducing clustering section. This variance in the weight multiplier will increase the variance in the length of ssion chains, which will also increase the variance of scores [START_REF] Sjenitzer | Variance reduction for xed-source Monte Carlo calculations in multiplying systems by improving chain-length statistics[END_REF].

Isotope Concentration

. Branchless collisions on the material

An alternative approach is to rst sample whether the reaction channel will be scattering or ssion, and then sample the isotope given this information. The probability of scattering and ssion are calculated following Eqs. (5.6) with the macroscopic material cross sections. If ssion was sampled, then the probability of sampling isotope i is

If scattering was sampled, then the probability of sampling isotope i is

.

(5.28)

The scattering channel α is then sampled with the same probability used in Eq. (5.25). The particle weight is modied using Eq. (5.5), using the macroscopic cross sections. Thus, in this sampling scheme, the weight multiplier is independent of the reaction channel and of the selected isotope. Should the sampled scattering channel not have a yield of 1, then we again multiply the particle weight by ν i,sα (E). Compared to the algorithm of the previous section, it is clear that this strategy requires larger modications to the algorithms already present in an existing Monte Carlo code.

. Numerical results

To illustrate the dierences between the two alternatives presented above, we shall now examine the case of a reected reactor lled with a homogeneous mixture of light water and uranium, the isotopic composition of which is provided in Table 5.6. This mixture is not intended to be realistic, but simply provide a multiplying system which we can use for our clustering study. Using the ENDF/B-VIII.0 library (with the thermal scattering law for light water), a 400 cm × 400 cm × 400 cm box of this composition is nearly critical.

As the box remains completely homogeneous, we are able to continue using Eq. (5.19) to approximate the ideal entropy for the box, given the number of particles and partitions. Figure 5.18 presents the entropy and the number of families for simulations, using dierent forms of branchless collisions, with either combing or splitting. It is immediately evident that, despite the box being nearly critical, branchless collisions do little to increase the saturation value of the entropy to the ideal value, unlike the case of single-speed transport (Fig. 5.1). Branchless collisions do seem to lead to a slightly higher entropy, with branchless collisions on the material faring only slightly better than branchless collisions on the isotope.

Looking at the number of particle families, we note that branchless collisions on the isotope lead to no signicant change with respect to branching collisions. Branchless collisions on the material do however seem to produce a slower decrease in the number of particle families. at.

On the contrary, it clearly appears that the system with L = 400 is strongly aected by spatial correlations. The convex shape arising from splitting events is amplied over time, and does not have time to relax between successive entropy steps. This behaviour can be further characterized by relying on previous insight gained on Brownian motion. The mixing time, as dened in Chapter 3 for a population of diusing neutrons in a one-dimensional system with mass-preserving boundary conditions, is given by τ p,D = 4L 2 /Dπ 2 . We can assume D = v/Σ t to recast the diusion-based parameters into a single-speed transport setting. For the system with L = 100, the mixing time is thus τ

p,D = 1.2178 s, while it is τ

(2) p,D = 19.48 s for the system with L = 400. By analogy with the analysis performed in Chapter 3 and 4, the ratio η = τ th /τ p,D predicts the strength of spatial correlations in the system. When η ≳ 1, spatial correlations will be washed between each entropy step (like in the L = 100 case), and the eect of splitting will be negligible. Otherwise, when η ≪ 1, spatial correlations will grow over successive entropy steps, and spatial clustering may occur (like in the L = 400 case).

In the subcritical system, the weight multiplier is smaller than one, and no splitting occurs. Spatial correlations cannot appear, and the Feynman moment will be at, with a value driven by the average neutron weight which will decrease in time, as illustrated by Fig. 6.3. For long times, the value of the Feynman moment will saturate at w R .

. The eect of combing on spatial correlations

Even in this simple system, the addition of weight combing has a drastic eect on the entropy, as illustrated in Fig. 6.4. It should be noted that our implementation of combing ensures that the number of Monte Carlo particles after combing is set to the integer closest to the total weight in the simulation, i.e. if the total weight of neutrons is W = 10000, then weight combing will comb the Monte Carlo particle population to N = 10000 neutrons of unit weight. Furthermore, neutron entropy and Feynman moments are computed just before combing is applied, i.e. they are aected by the weight uctuations within a time step. The ideal entropy in Fig. 6.4 is computed for the total neutron weight at time t, rounded to the closest integer. Combing is applied on the same time grid as the one used to score time-dependent tallies, with δt = 0.02 s.

In the supercritical conguration, the entropy steps disappear, but additional correlations occur in the simulation, as illustrated by the discrepancy between the neutron entropy and the ideal entropy for the corresponding (now strictly increasing) number of Monte Carlo particles. In the system with L = 100, the Monte Carlo entropy appears to converge towards the ideal entropy for large times, whereas the discrepancy between the Monte Carlo and ideal entropy remains for the system with L = 400 ; for longer times, we checked that the discrepancy disappears even in the large system, albeit slowly. For both boundary to T = 10 s (and δt = 0.1 s) in order to observe at least one oscillation. In the absence of combing, we observe the usual jump at short times, as in the supercritical and subcritical cases ; we also observe the oscillations, which suggests that the critical regime is not `special'. We veried that, for (very) long times, the entropy oscillations are dampened, and the entropy converges towards an asymptotic value. When applying combing, the only dierence with respect to the non-stationary states is that the entropy converges more rapidly towards an asymptotic value. We also veried that the behaviour of the Feynman moments is qualitatively similar to the one of the supercritical system (not shown here for conciseness).

. The eect of precursors on spatial correlations

We illustrate now the eect of adding precursors to the neutron model of the previous section on spatial correlations, using either analog or forced decay, which are described in Sec. 2.4.2. For this purpose, we introduce a delayed-neutron precursor family in the system with leakage, assuming that we apply combing at each time step. Let the single precursor family be characterized by a delayed fraction β = 0.1 and a decay constant λ = 0.01 s -1 . The parameters of the precursor family are not realistic, but they are chosen to speed up the simulation times, yet retaining a sucient separation between the time scales of neutrons and precursors. The introduction of the combined precursor (which requires several precursor families) is left for Sec. 6.3.

When using forced decay, the weight of the delayed neutrons produced in each time step is proportional to the time step size ; a small time step allows investigating the fast dynamics of prompt neutrons, but leads to a large number of neutrons of small weight. In contrast, a larger time step makes it harder to follow the time evolution of the prompt neutron population. Note that this last remark could be mitigated using the relative importance introduced by Faucher et al., and increasing the importance of neutrons compared to precursors [START_REF] Faucher | New kinetic simulation capabilities for Tripoli-4®: Methods and applications[END_REF]. If a neutron is produced through forced decay with a statistical weight w < w R , it is immediately tested for Russian roulette (i.e. we do not wait for the rst collision).

We will investigate supercritical, subcritical and critical congurations. We take the nal time to be T = 200 s, which is of the order of 1/λ ; with such a long nal time, the simulations of the promptsupercritical systems of Tab. 6.1 would be prohibitively costly. Therefore, we use instead the ssion yields given in Tab. 6.2, which yield prompt-subcritical systems.

Analysis of correlations in the equilibrium source

The sampling of the equilibrium source is independent of the choice of the decay algorithm in the time-dependent simulation. The Feynman moments of the equilibrium source, before and after combing, are presented in Fig. 6.12 for the supercritical congurations. We notice that the Feynman moment of precursor, its statistical weight can only decrease (in the case of forced decay) or be constant (in the case of analog decay). Thus, assuming that a neutron started with unit weight (i.e., from combing at the end of the previous time step), the maximum expected weight for a neutron at the end of the time step is again given by w δt = f δtvΣt ; this leads to w δt = 1.82 in the small system, and w δt = 1.11 in the large system. The value observed in Figs. [START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF].15a and 6.15b are relatively close to the expected neutron weight at the end of a time step, and we can conclude the spatial uctuations of the neutron population are close to being Poisson distributed. The Feynman moments for the neutron population behave similarly when using either forced or analog decay ; therefore, we only show the Feynman moments with forced decay. Note that, for long times, the spatial shape of the Feynman moment for neutrons becomes slightly convex.

Figure 6.14b for the small system indicates a qualitatively dierent behaviour for the precursor entropy when using forced or analog decay. In the analog case, entropy rst decreases, suggesting that spatial correlations originating from the equilibrium source rst favour spatial clustering in the time-dependent process. However, when approaching t = 1/λ, most of the initial precursor population has decayed, and the spatial correlations in the precursor population start to be driven by the time-dependent process.

The ensuing linear growth corresponds to the exponential increase of the precursor population. This is illustrated by Fig. 6.16a representing the Feynman moment for the precursor population when using (a) For the small system.

(b) For the large system Figure 6.17 : Feynman moment of the precursor population in the supercritical case for N 0 = 1000, for different times. We use forced decay.

analog decay : around t = 50 s, the Feynman moment forgets the spatial shape inherited from the equilibrium source. On the other hand, it is less clear why we do not observe such behaviour when using forced decay ; we conjecture that it is due to the entropy being related to the number of Monte Carlo particles. With forced decay, the number of precursor particles is increasing within a time step. Since we compute entropy before combing is applied, a higher number of precursors induces a higher entropy.

While the time needed for the precursor population to forget the correlations of the equilibrium source is similar using forced decay or analog decay (as illustrated in Fig. 6.17a), the behaviour of the entropy is mainly driven by the number of Monte Carlo particles. It is interesting to note that the value of the Feynman moment appears to decrease with time when using forced decay. The decrease of the average precursor weight within a time step is given by w δt ∼ e -λδt ≃ 0.995 for this system. This value does not explain the ones observed in Fig. 6.17a, and we have not explanation of why the Feynman moment of the precursor population decreases with time.

On the other hand, Fig 6 .14b indicates that, in the large system, the precursor entropy behaves similarly for forced and analog decay. The entropy when using forced decay is larger, due to the fact that precursor particles are not destroyed upon decay. However, the analysis of the Feynman moment of the precursor population in the large system indicates a signicant dierence between using forced or analog decay. Figure 6.16b shows that the value of the Feynman moment for analog decay is close to the expected precursor weight at the end of the time step, indicating that spatial uctuations are almost Poisson distributed. However, Fig. 6.17b shows that the value of the Feynman moment steadily decreases with time, until it cannot be related to the average precursor weight anymore ; this suggests that spatial uctuations in the precursor population are quenched over time. We did not nd a plausible argument to explain this behaviour.

Analysis of correlations in the subcritical case

The time-evolution of the entropy in the subcritical case is presented in Fig. 6.18. For the neutron entropy, the comments in the supercritical cases hold true. The behaviour of the precursor entropy, however, diers signicantly. When using analog decay, the precursor entropy decreases with time, which encodes the fact that the neutron population is decreasing over time. However, when using forced decay, the entropy rst increases until around t ∼ 1/λ, before it starts decreasing, which is reminiscent of Fig. 6.14b ; the reason is, again, that 1/λ is the typical time it takes for precursors stemming from the equilibrium source to decay, and thus it is also the typical time it takes for precursors to forget the correlations of the initial distribution.

Again, the Feynman moments of the neutron population are independent of the use of forced or analog decay ; therefore, we only show those with forced decay. In the small system (see Fig. 6.19a), the slightly less sensitive to statistical uctuations, and thus to clustering. On the other hand, Fig. 6.22b resembles Fig. 6.14b for the supercritical case, where the linear increase for long times (which corresponds to the exponential population growth) is replaced by an asymptotic convergence to a constant value (which corresponds to the constant population).

The Feynman moments of the neutron population lead to the same conclusions as in the supercritical case (see Fig. 6.15), and are not shown here for conciseness. The Feynman moment of the neutron population is increasing with time and saturates at a value around w δt ≃ 1.72 for the small system, and w δt ≃ 1.04 for the large system. The Feynman moment of the precursor population is given in Figs. [START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF].23 and 6.24, and qualitatively resembles that of the supercritical population in Figs. [START_REF] Caron | New aspects in the implementation of the quasi-static method for the solution of neutron diusion problems in the framework of a nodal method[END_REF].16 and 6.17.

(a) Case of the small system.

(b) Case of the large system Figure 6.24 : Feynman moment of the precursor population in the critical case for N 0 = 1000, for different times. We use analog decay. The reactor is assumed to be critical at the initial state (t = 0) ; for the time-dependent simulations, we will consider the case where the reactor stays critical, and the case where a certain amount of reactivity is introduced at t = 0 : this latter conguration is achieved by modifying the capture cross sections. The behaviour of space-time correlations in the critical and non-critical regimes will be investigated by using the Shannon entropy and the Feynman moments for the neutron and precursor populations as dened in Sec. 6.1. We refer the reader to Chapter 5 for a thorough description of these observables.

. Simulation parameters for the homogeneous benchmark

We briey recall the features of the homogeneous benchmark introduced in Chapter 5. We consider two system sizes, L = 50 (system 1) and L = 100 (system 2) of a homogeneous slab geometry with leakage boundary conditions. The nuclear data are represented in a three-group formalism for neutrons, as recalled in Tabs. 5.2 and 5.5. The nuclear data for the reactivity insertions and withdrawals is given in Tabs. 6.4 and 6.5. The reactivity insertions δρ = 1/k e -1/k t≥0 are provided in Tab. 6.6, where k e is the multiplication factor in the critical cases 1 . Note that our choice of reactivity insertion does not make the system prompt-supercritical. Since we now explicitly consider precursors, the ssion spectrum given in Tab. 5.2 needs to be adjusted, and is given in Tab. 6.4. The six-family nuclear data for precursors are taken from Vitali et al. [START_REF] Vitali | Eigenvalue separation and eigenmode analysis by matrix-lling Monte Carlo methods[END_REF] (inspired by those of 235 U), and are provided in Tab. 6.2. Space is partitioned into intervals of equal length δx = 0.5 cm, and we chose a time step δt = 1 ms. Note that the slowest precursor decay typical time is 1/λ 1 ∼ 100 s : transients on the space-time correlations within the precursor population might thus last well beyond 100 s ; due to the involved computational burden, for our simulations will nonetheless set a nal time T = 10 s. entropy increases in the supercritical case, which is related to the increasing population size. Conversely, the neutron entropy decreases in the subcritical case, which is related to decreasing population size. In this case, the long-time variations of the neutron entropy cannot be used to infer the emergence of spatial correlations in these systems.

We turn to the precursor entropy in Fig. 6.31b. The behaviour of the precursor entropy in the supercritical cases is very similar in both systems, with a monotonic increase with time. In the subcritical case, although both systems still behave similarly, we observe that a maximum is attained near t ≃ 4.5 s, and then the entropy decreases. The maximum can safely be associated to the fact that spatial correlations stemming from the kinetic sources progressively relax, while the subsequent decreases can be associated to the decreasing population size. In any case, the interpretation of the observed behaviour in terms of spatial correlations is not apparent.

We shall now rely on the Feynman moments to clarify our previous discussion. The Feynman moments of the neutron population lead to the same conclusions as in the critical case, and will therefore not be shown here. Figure 6.32 indicates that the prompt jump has a strong inuence on the Feynman moments of the collision counts, which then saturate in the long-time behaviour. It is however unclear whether this conclusion would still hold if the nal time of the simulation were longer. This behaviour is reminiscent of the fact that, in power iteration calculations, the Feynman moments are independent of the population correlations is marginal. Notably, the critical catastrophe is eectively averted by the use of non-analog methods.

-Conclusions and perspectives

Thanks to the combined benets of the improvements in the algorithms and of the increased computer power, it becomes possible today to address non-stationary problems in reactor physics by Monte Carlo simulation, which allows for an `exact' treatment of the geometry and of the nuclear data of the simulated system, introducing virtually no approximation nor discretization eects. In view of the intrinsically stochastic nature of the sampled events along each particle history, Monte Carlo tallies uctuate, and the estimates of the average quantities of interest come with a statistical uncertainty that is asymptotically inversely proportional to the square root of the number of histories. The almost complete absence of approximations is thus thwarted by the very high calculation cost required to achieve the target accuracy, especially in kinetic (i.e. time-dependent) simulations. Furthermore, the simulated particle histories in kinetic problems are correlated, due to ssion events, which makes the behaviour of the statistical uncertainty rather complex : the eects of correlations might for instance manifest themselves in the form of particle clustering, the spatial distribution of the simulated particles being extremely `patchy', instead of uniform as expected. Since in most cases Monte Carlo simulations involve some forms of nonanalog sampling, usually in combination with variance-reduction and population-control techniques, the evolution of uctuations and correlations in kinetic Monte Carlo simulations depends on the non-trivial interplay between the way the sampling strategies are conceived, and the eects inherent to the physicsinspired underlying stochastic processes. An overview of these issues has been presented in Chapter 1.

Prompted by these considerations, in this thesis we have developed a unied and coherent framework for the analysis of uctuations and correlations in kinetic Monte Carlo simulations, for the purpose of ensuring the reliability of the statistical uncertainty estimates in this class of calculations. In particular, we have focused on the interplay between the behaviour of uctuations and correlations and the application of variance-reduction and population-control techniques, as well as sampling strategies used for nonanalog Monte Carlo simulations. Our investigation has covered both algorithms devoted specically to time-dependent problems and algorithms devoted to (stationary) eigenvalue problems, which are key for the sampling of the equilibrium neutron and precursor population.

Before presenting our novel contributions, in Chapter 2 we have recalled the general framework for both the eigenvalue and time-dependent formulation of the Boltzmann equation. For the time-dependent formulation, we have emphasized the crucial role of the delayed neutron precursor contributions. Correspondingly, we have then summarized the state-of-the-art of non-analog Monte Carlo strategies for reactor physics problems, with special emphasis on the methods used for kinetic problems. Monte Carlo simulations produce unbiased estimates of the rst moments of the sought observables, which are precisely the solutions of the Boltzmann equation. We have discussed the main sampling algorithms for particle transport, and illustrated a few estimators that can be used in order to estimate the observables of interest. Furthermore, we have presented an overview of the most important variance-reduction and population-control schemes that can help reduce the statistical uncertainty and prevent the size of the particle population from becoming too large or too small. In addition, we have introduced simplied formulations of the Boltzmann equations, encompassing the multi-group formalism and the diusion approximation, and we have shown how these equations can be also solved by Monte Carlo methods, by sampling the random walks of neutrons and precursors in the phase space.

For the investigations discussed in this thesis, we have developed two simplied Monte Carlo miniapps, also presented in Chapter 2. The main argument behind this choice, as opposed to adding new functionalities and estimators to TRIPOLI-4 ® , the production-level Monte Carlo code developed at CEA, is that our analysis requires tools such as two-point correlation functions, which are typically very expensive to compute and memory-greedy. Furthermore, we often needed to extensively modify the tradi- De plus, certains résultats ont été présentés en conférence sous la forme de présentations orales :

• International Conference on Transport Theory (ICTT), Bertinoro, Italy, Jul. 10-16th, 2022.

• M&C 2023 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Niagara Falls, Canada, Aug. 13-17th, 2023.

• Quatrième workshop "Mathématiques pour la neutronique", GdR MaNu, INSTN/Saclay, France, Dec. 7th, 2022.

• Présentation orale invitée à la première conférence du GdR Branchement, Institut de Mathématiques de Toulouse, France, Nov. 8-10th, 2023.