Jean-Yves Fortin
email: jean-yves.fortin@univ-lorraine.fr

Xavier Durang

Mooyoung Choi
email: mychoi@snu.ac.kr

Limited coagulation-diffusion dynamics in inflating spaces

Keywords: numbers: 05.10.Gg, 05.40.-a, 05.40.Fb

We consider the one-dimensional coagulation-diffusion problem on a dynamical expanding linear lattice, in which the effect of the coagulation process is balanced by the dilatation of the distance between the particles. Distances x(t) follow the general law ẋ(t)/x(t) = α(1 + αt/β) -1 with growth rate α and exponent β, describing both algebraic and exponential (β = ∞) growths. In the space continuous limit, the particle dynamics is known to be subdiffusive, with the diffusive length varying like t 1/2-β for β < 1/2, logarithmic for β = 1/2, and reaching a finite value for all β > 1/2. We interpret and characterize quantitatively this phenomena as a second order phase transition between an absorbing state and a localized state where particles are not reactive. We furthermore investigate the case when space is discrete and use a generating function method to solve the time differential equation associated with the survival probability. This model is then compared with models of growth on geometrically constrained two-dimensional domains, and with the theory of fractional diffusion in the subdiffusive case. We found in particular a duality relation between the diffusive lengths in the inflating space and the fractional theory.

Introduction

Cooperative effects in low-dimensional systems with strongly interacting particles present a rich variety of critical properties in the regime out of equilibrium. Model examples of such systems are provided by one dimensional reaction-diffusion processes [START_REF] Ben Avraham | Diffusion and reactions in fractals and disordered systems[END_REF][START_REF] Henkel | Non-equilibrium phase transitions: absorbing phase transitions[END_REF][START_REF] Ódor | Universality in non-equilibrium lattice systems[END_REF], which are relevant to a number of non-equilibrium physical cases, such as excitons in polymer chains TMMC=(CH 3 ) 4 N(MnCl 3 ) [START_REF] Kroon | Diffusion-limited exciton fusion reaction in onedimensional tetramethylammonium manganese trichloride (tmmc)[END_REF] or relaxation of photoexcitations in carbon-nanotubes [START_REF] Russo | Onedimensional diffusion-limited relaxation of photoexcitations in suspensions of single-walled carbon nanotubes[END_REF]. The low dimensionality induces strong fluctuations that dominate the kinetics, and invalidates all descriptions based on mean-field theories below the upper critical dimension d c = 2. Exact results are therefore of importance in describing the dynamical behavior with precision by taking into account all the correlations.

Here we consider one of these coagulation-diffusion processes on a one-dimensional lattice with discrete sites of elementary size a. The Markov dynamics is defined by the particle elementary moves at the same rate τ -1 : A + ∅ → ∅ + A or ∅ + A → A + ∅ for the diffusion process, and A + A → A + ∅ or A + A → ∅ + A for the irreversible coalescence process with probability one. It is known that this out of equilibrium dynamics tends to decrease the concentration of particles with time like t -1/2 [START_REF] Spouge | Exact solutions for a diffusion-reaction process in one dimension[END_REF][START_REF] Ben Avraham | Diffusion and reactions in fractals and disordered systems[END_REF], instead of t -1 in the mean field description, hence driving the system to an absorbing state consisting of one particle.

External sources such as particle input [START_REF] Doering | Diffusion-limited coagulation in the presence of particle input: Exact results in one dimension[END_REF][START_REF] Hinrichsen | Universality properties of the stationary states in the one-dimensional coagulation-diffusion model with external particle input[END_REF] leads on the contrary to an equilibrium state, and the model can be solved using for example the eigenvalue formalism. The opposite case of a source of particles would be insertion of empty spaces, which can be modeled by expanding the space or the geometry uniformly and independently of the presence of particles, modifying the Brownian dynamics and therefore the Fokker-Planck equation [START_REF] Ali | Scale-invariant growth processes in expanding space[END_REF][START_REF] Yuste | Diffusion in an expanding medium: Fokker-Planck equation, Green's function, and first-passage properties[END_REF][START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF]. This has many practical applications for example in the physics of bubble coalescence in expanding foams [START_REF] Bhakta | Decay of standing foams: drainage, coalescence and collapse[END_REF][START_REF] Mancini | Structure and Evolution of Soap-Like Foams[END_REF], or gravitational merging of galaxies or masses as well as ray diffusion in an inflating universe [START_REF] Berezinsky | Diffusion of Cosmic Rays in the Expanding Universe[END_REF]. Expanding media also have a strong influence on the dynamics of cells induced by tissue growth in biology [START_REF] Ali | Reproduction-time statistics and segregation patterns in growing populations[END_REF][START_REF] Yates | Discrete and continuous models for tissue growth and shrinkage[END_REF][START_REF] Matthew | Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization[END_REF]. In particular, geometrical constraints in two-dimensional competitive environments with gene segregation for two different kinds of bacteria [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF][START_REF] Nadell | Spatial structure, cooperation and competition in biofilms[END_REF], such as Eschrichia coli, exert also an influence on the growth of the different colonies when it is directed radially in presence of decimation at the boundaries between two different colonies [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF][START_REF] Ali | Reproduction-time statistics and segregation patterns in growing populations[END_REF]. In this radial configuration, the frontiers or domain boundaries between two competitive colonies display superdiffusive spatial fluctuations with a wandering exponent 0.65 > 1/2, greater than the exponent of the simple Brownian diffusion.

In this paper we investigate the effect of geometrical time expansion for Brownian particles with coalescence. We review in section 2 the general results about the diffusion properties of a single particle using a discrete analysis (discrete space and time), and in the continuous limit identifying two different regimes corresponding to subdiffusion and localization.

Section 3 is dedicated to describing analytically the dynamics of many particles with coalescence through diffusion, using the results of the previous section obtained in the literature and the empty interval method in the continuous case. We identify the two regimes as a second order phase transition, using the number of particles as the order parameter and the expansion exponent as the inverse of a temperature. In particular, we can show the model presents slowing down characteristics at the transition.

In section 4, we develop an algorithm for the stochastic process in an expanding medium, using a combination of continuous and discrete methods for the space and particle dynamics respectively. The results confirm the analytical and scaling expressions for the particle concentration of section 3 in the different regimes.

In section 5, we extend the solution in the case where space is discrete and time continuous. We propose two methods to solve the corresponding diffusion equation for the empty interval distribution relevant for various physical systems, and compare the solutions with the numerical results of section 4.

In section 6, we discuss the relevance and connection of the model with the problem of particle diffusion in constrained two-dimensional geometries, including radial constraints and generalized Brownian noise.

Finally, in section 7, we consider the model in the more general context of abnormal diffusion described by the action of fractional operators. The aim is to examine the correspondence and analogy between the problem of diffusion in expanding space and the formalism of diffusion with adequate fractional operators, involving continuous time random walk (CTRW) models. We shall conclude from this analysis of the differences and relevance of the fractional formalism. We establish in particular a duality relation between diffusive lengths which holds for all the regimes.

Effective diffusion length

To define more precisely the problem, we consider a lattice initially filled randomly with particles at a given concentration c 0 , and we constraint the lengths between any two spatial points of the lattice to increase with time at a certain rate (or by adding for example empty sites on a discrete chain uniformly at random locations), with distances growing with time like a power or exponential law. The space inflation tends to increase distances between the particles, preventing them to coagulate by diffusive motion. It is expected that the two opposite contributions (diffusive motion and space inflation) will induce a transition between two regimes. To analyze in detail this phenomenon, let assume that all distances x(t) increase according to the general law

x(t ′ ) x(t) = 1 + αt ′ /β 1 + αt/β β ( 1
)
where β is a given exponent and α is the growth rate of the lattice, or the amount of new empty spaces added uniformly to the lattice per unit of time. The dynamics of the lattice modifies the diffusive length of particles. Indeed, we can evaluate the diffusive length in this system by considering the stochastic motion of a single particle: If we define the growth factor r(t) = (1 + αt/β) β , and x k the position of the particle at time t k = kτ , where τ is the unit of time, we can write the discrete equation of motion as

x k = r k r k-1 (x k-1 + ξ k ) (2)
where x 0 = 0, r k = r(t k ), and ξ k a Gaussian white noise satisfying ξ k = 0 and

ξ k ξ k ′ = 2a 2 δ k,k ′ .
Solving the previous equation leads directly to

x k = r k r k-1 ξ k + r k r k-2 ξ k-2 + • • • + r k r 0 ξ 1 . (3)
The typical mean squared displacement of the random walk x 2 k is found to be equal to

x 2 k = 2a 2 r 2 k 1 + 1 r 2 1 + • • • + 1 r 2 k-1 = 2a 2 r 2 k k-1 j=0 1 (1 + ατ j/β) 2β (4)
from which we can extract an effective or comoving diffusive length ℓ(t k ) = x 2 k 1/2 /r k whose long time behavior depends on the value of β. We first define the continuum limit a → 0 and τ → 0, such that the diffusive constant D = a 2 /τ remains finite, which gives [START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF]

ℓ 2 (t) = 2D β α(1 -2β) (1 + αt/β) 1-2β -1 . (5)
If 0 ≤ β < 1/2, we have ℓ(t) ∼ t 1/2-β , the effective diffusive length increases but less than the usual ℓ(t) = √ 2Dt 1/2 Brownian motion length and the process is subdiffusive [START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF][START_REF] Yuste | Subdiffusion-limited A + A reactions[END_REF][START_REF] Hyung Jeon | Scaled Brownian motion: A paradoxical process with a time dependent diffusivity for the description of anomalous diffusion[END_REF]. If β = 1/2, the length diverges logarithmically ℓ 2 (t) ∼ (D/α) log(1 + 2αt), and if β > 1/2, it reaches a finite value ℓ 2 (t) ≃ 2Dβ/α(2β -1) ‡. In particular, in the case of an exponential growth, β = ∞, ℓ 2 (t) = D(1e -2αt )/α ≃ D/α. When β = 0 and/or α = 0, we recover the usual Brownian law ℓ(t) ∼ t 1/2 . We therefore expect from this analysis that the coagulation-reaction process will stop when β > 1/2 after a transit time since effectively the motion of the particles will be frozen. This limit is similar to a dynamical localization of the particle, as its motion is impeded by the space expansion.

Empty interval distribution method for the particle concentration

In order to find the time dependence of the particle concentration, we need to solve the reaction-diffusion problem. It is convenient to use the empty interval method based on the interparticle distribution function (IPDF) [START_REF] Daniel Ben | Complete exact solution of diffusion-limited coalescence, A + A → A[END_REF], with which the problem becomes linear and solvable: We define the probability to have at least n consecutive empty spaces • by

E(n, t) = Prob(× • • • • • n ×) (6)
where × represents equivalently either a particle • or an empty space •. As before, we assume that τ -1 is the rate of diffusion to the right or left at a given time step. [START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF], the corresponding parameters are the following:

ℓ 2 (t) → 2Dτ (t), β → γ, β/α → t 0 and α → H
The operator n∂ n is the usual dilatation operator and appears naturally in this equation. To solve equation ( 8), we consider the continuum limit of the space variable as follows.

The discrete case will be investigated in section 5. In the limit a → 0, we introduce continuous space coordinates x = na and diffusion coefficient D = a 2 /τ . The time evolution of the IPDF then becomes

∂E(x, t) ∂t = 2D ∂ 2 E(x, t) ∂x 2 - αx 1 + αt/β ∂E(x, t) ∂x . (9)
It is convenient to perform the transformations y = xr -1 (t) as the new comoving space variable, and u = u(t) as new time variable. This change of variables simplifies the previous equation which becomes

∂E(y, u) ∂u = 2D ∂ 2 E(y, u) ∂ 2 y ( 10
)
where E(y, u) = E(x, t), provided that u(t) satisfies

u(t) = r -2 (t), u(0) = 0 u(t) = β α(1 -2β) (1 + αt/β) 1-2β -1 = ℓ 2 (t) 2D . (11)
We can therefore relate naturally the proper time u(t) to the effective diffusive length defined in equation [START_REF] Russo | Onedimensional diffusion-limited relaxation of photoexcitations in suspensions of single-walled carbon nanotubes[END_REF]. Equation ( 10) is the usual diffusion equation. The Laplace transform Ê(y, s) with respect to the time variable u verifies the differential equation

s Ê(y, s) -E(y, 0) = 2D ∂ 2 Ê(y, s) ∂ 2 y ( 12
)
where E(y, 0) = E(y, 0) is the initial condition for the interval distribution, with y = x at t = 0. We will consider in the following an initial exponential distribution E(y = x, 0) = e -c 0 x , for which the average particle density is c 0 . The solution of this differential equation is simply given by two exponential functions, in addition to the particular solution associated to E(y, 0)

Ê(y, s) = a(s) exp - s 2D y + b(s) exp s 2D y + 1 √ 8Ds exp - s 2D y y 0 dy ′ exp s 2D y ′ E(y ′ , 0) -exp s 2D y y 0 dy ′ exp - s 2D y ′ E(y ′ , 0) . ( 13
)
Coefficients a(s) and b(s) depend only on s. They are determined by the two conditions:

E(0, u) = 1, lim y→∞ E(y, u) = 0. ( 14
)
The first one implies Ê(0, s) = 1/s, and fixes the value of a(s) + b(s) = 1/s. Since Ê(y, s) should remain finite when y → ∞, the second condition restricts b(s) to be equal to

b(s) = 1 √ 8Ds ∞ 0 dy ′ exp - s 2D y ′ E(y ′ , 0). ( 15
)
We then use the following inverse Laplace functions when x > 0 [24]

L -1 1 s exp - s 2D y = erfc y √ 8Du , L -1 1 √ s exp - s 2D y = 1 √ πu exp - y 2 8Du ( 16
)
to obtain the general expression of E(y, u)

E(y, u) = erfc y √ 8Du + 1 √ 8Dπu ∞ 0 dy ′ exp - (y -y ′ ) 2 8Du -exp - (y + y ′ ) 2 8Du E(y ′ , 0). ( 17
)
The particle concentration is determined by the relation

c(t) = -∂ x E(x, t)| x=0 = -r -1 (t)∂ y E(y, u)| y=0
, and the number of particles as function of time for a system initially of size L 0 is then equal to

N p (t) = c(t)L 0 r(t) = -L 0 ∂ y E(y, u)| y=0 .
If the initial number of particles is N 0 = c 0 L 0 , then the ratio N p (t)/N 0 is equal to, for an initial exponential distribution of empty intervals, see also [START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF] N p (t)

N 0 = e c 2 0 2Du(t) erfc c 0 2Du(t) = e c 2 0 ℓ 2 (t) erfc (c 0 ℓ(t)) (18)
where ℓ(t) is the effective diffusive length given by equation [START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF]. When β < 1/2, ℓ(t) ≃ t 1/2-β and the number of particle decreases like a power law N p (t)/N 0 ≃ ( √ πc 0 ℓ(t)) -1 ∼ t -ν , with exponent ν = 1/2β. When β = 1/2, the number of particle decreases logarithmically like log(t) -1/2 . In the case β > 1/2, the number of particles, after a regime of coalescence where it decreases, reaches a finite limit given by the dimensionless parameter λ defined in the following

N p (∞) N 0 = e λ 2 erfc(λ), λ = 2βD α(2β -1) c 0 ≃ 1 √ πλ 1 - 1 2λ 2 , λ ≫ 1 ≃ 1 - 2λ √ π , λ ≪ 1. ( 19
)
For short times, αt ≪ 1, the coagulation process is not affected by the space expansion since ℓ(t) ≃ √ 2Dt. The result in equation ( 18) is similar to the concentration of the purely diffusive-coalescence model (α = 0) as function of time [START_REF] Durang | Exact correlations in the one-dimensional coagulation-diffusion process investigated by the empty-interval method[END_REF]

c(t) = c 0 e c 2 0 ℓ 0 2 (t) erfc(c 0 ℓ 0 (t)) ≃ (2πDt) -1/2 (20)
where ℓ 0 (t) = √ 2Dt replaces ℓ(t) instead. Identifying this expression with equation ( 18), we can evidence the crossover regime of dissociation at which the coagulation process due to the diffusive motion stops at a characteristic time defined by ℓ 0 (t)c 0 = λ, and which is equal to

t c = β/[α(2β -1)].
It is interesting to make an analogy with phase transition phenomena. Indeed we can consider N p /N 0 as an order parameter and β as the inverse of a temperature, with β c = 1/2 the critical point. In the steady state limit, the previous results can be summarized as follows:

N p (∞) N 0 = 0, β < β c N p (∞) N 0 ≃ α πDc 2 0 1 -β c /β, β β c . (21)
This corresponds, by identification, to a second order phase transition, between an absorbing state (high temperature) and a localized state of particles (low temperature) where N p is the number of localized particles. In terms of electronic localization, the critical point may describe the transition between fully delocalized electrons and partially localized electrons, β being a parameter similar to the strength of a random potential on a one-dimensional lattice, or inverse temperature. The critical slowing down phenomenon at β = β c is evidenced by the fact that the effective diffusive length is diverging logarithmically. We can also notice that the remaining number of particles in inversely proportional to the concentration since the coagulation reactions are more effective when more particles are initially closer to each other, before the dissociation time occurs.

Numerical simulations

The analytical result given by equation ( 18) is compared to numerical simulations, first in the subdiffusive case β ≤ 1/2, see figures 1 and 2. N 0 particles are randomly distributed on a chain of size L 0 with an intrinsic unit step size ∆x. At each time step ∆t, a site of width ∆x is randomly chosen. If it is occupied by a particle, this one diffuses by one step ∆x on the right or left with equal probability. If the destination site contains another particle within the distance ∆x, the two particles coalesce. After this process, the size of the system, as well as the distances between the remaining particles, are increased by a factor r(t + ∆t)/r(t). The number of particles is then evaluated after 10 7 time steps by averaging over a set of random initial samples. The evolution of the number of particles is displayed in figure 1 for an initial system size L 0 = 1000 and c 0 = N 0 /L 0 = 1. In the inset, the approximate power law exponent ν is evaluated and compared to the exact expression ν = 1/2β. For comparison, in figure 2 is plotted the function given by equation [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF] for several values of β ≤ 1/2. In the exponential case (β = ∞), we have plotted in figure 3 the numerical value of the ratio of particles for a linear chain of L 0 = 1000 sites and different values of α, as function of the dimensionless time variable αt, and for an initial concentration c 0 = 1. In inset is displayed the final value of the remaining number of particles, as function of λ = c 0 (D/α) 1/2 , with the diffusive constant we have estimated to be equal to D ≃ (8N 0 ) -1/2 in dimensionless units. This gives a good agreement with the numerical data. The analytical solution, equation [START_REF] Nadell | Spatial structure, cooperation and competition in biofilms[END_REF], is represented in figure 4 for different values of λ and α = 10 -4 . The red dashed curve represents the time evolution of the particle number in absence of expansion, which shows approximately where the dissociation time t c is located.

Discrete solution for the IPDF

In this section, we investigate the solutions of the linear partial differential equation [START_REF] Hinrichsen | Universality properties of the stationary states in the one-dimensional coagulation-diffusion model with external particle input[END_REF], when the time is continuous but space discrete, using generating functions. Mathematically, similar discrete equations have been solved using Poisson-Jacobi transform, see for example reference [START_REF] Pathak | Poisson-Jacobi transforms[END_REF], but only in the case where the coefficients are independent of time. We start by writing equation ( 8) in a symmetric form

τ ∂E(n, t) ∂t = 2E(n + 1, t) -4E(n, t) + 2E(n -1, t) - ατ /2 1 + αt/β n (E(n + 1, t) -E(n -1, t)) (22)
with constraint E(0, t) = 1. We first notice that extending the interval length n to negative values is possible using the symmetry E(-n, t) = 2 -E(n, t), see reference [START_REF] Durang | Exact correlations in the one-dimensional coagulation-diffusion process investigated by the empty-interval method[END_REF]. This relation can be found easily in the pure diffusive case when α = 0 but also holds here for equation [START_REF] Hyung Jeon | Scaled Brownian motion: A paradoxical process with a time dependent diffusivity for the description of anomalous diffusion[END_REF] and can be proved by recurrence. To solve the case α = 0, we consider the generating function G(z, t) = n E(n, t)z n . The differential equation satisfied by G(z, t) is therefore

τ ∂G(z, t) ∂t = (2 + γ(t)) z + 1 z -4 G(z, t) -γ(t)(1 -z 2 ) ∂G(z, t) ∂z , γ(t) = ατ /2 1 + αt/β = τ 2 ṙ(t) r(t) . ( 23
)
It is important to note that this equation is invariant by transformation z → 1/z. A convenient way to reduce the previous equation is to use the change of variable

y = r(t)(1 -z)/(1 + z), for which the new function G(y, t) = G(z, t) satisfies τ ∂ G(y, t) ∂t = (2 + γ(t)) r(t) -y r(t) + y + r(t) + y r(t) -y -4 G(y, t) (24)
the solution of which is

G(y, t) = F (y) exp - 4t τ + t 0 dt ′ τ (2 + γ(t ′ )) r(t ′ ) -y r(t ′ ) + y + r(t ′ ) + y r(t ′ ) -y . (25)
Therefore, we obtain after some algebra and using the variable = F (wr(t))

w = (1 -z)/(1 + z) G(z, t) = r(t)F (wr(t)) exp 4 t 0 dt ′ τ (2 + γ(t ′ )) ∞ k≥1 w 2k r 2k (t) r 2k (t ′ ) 10 - 7 10 -6 10 -5 10 -4 10 -3 10 -2 10
(1 -w 2 )r(t) 1 -w 2 r 2 (t) exp 8β ατ ∞ k=1 w 2k 2βk -1 r 2k (t) -1 - αt β . (26)
The sum over k is related in the general case for β to Lerch transcendent function

k≥1 z k 2βk -1 = z 2β Φ z, 1, 1 - 1 2β .
In the limit β = 0 or r(t) = 1, after approximating the factor r 2k (t) ≃ 1 + 2kαt, we recover the generating function for the diffusive case in terms of modified Bessel functions

G(z, t) = F (w) exp - 4t τ + 2 z + 1 z t τ = G(z, 0) exp - 4t τ n I n 4t τ z n (27)
since F (w) = G(z, 0) = m E(m, 0)z m . We obtain, by identification of the power z n in equation ( 27) and using the symmetry relation E(-n, t) = 2 -E(n, t), the following expansion, generalizing previous results [START_REF] Spouge | Exact solutions for a diffusion-reaction process in one dimension[END_REF] = e -4t/τ m≥1 E(m, 0)

E(n, t) = e -4t/τ m E(m, 0)I n-m 4t τ (28) 10 -6 10 -5 10 -4 10 -3 10 -2 10
I n-m 4t τ -I n+m 4t τ + 2 m≥1 I n+m 4t τ + I n 4t τ .
If we start with a lattice filled with one particle per site, E(m ≥ 1, 0) = 0, the previous equation is simplified as only the last two terms remain. The particle concentration is given by c(t) = 1 -E(1, t) and therefore we obtain for the non-expanding lattice case

c(t) = e -4t/τ I 0 4t τ + I 1 4t τ ≃ τ 2πt . ( 29
)
Another way to recover this result in the large time limit and for the same initial conditions is to use the complex integral with a path around the unit circle to obtain the value of E(n, t)

E(n, t) = dz 2iπz n+1 G(z, t) (30)
= dz 2iπz n+1 1 + 2 m≥1 z -m exp - 4t τ + 2 z + 1 z t τ .
If we set z = e iθ we obtain

E(n, t) = π -π dθ 2π e -inθ + 2 m≥1 e -i(m+n)θ exp - 4t τ (1 -cos θ) .
For t large, we can develop the cosine term for θ small, and integrate over θ the resulting Gaussian integral after extending the range of integration to infinity

E(n, t) ≃ τ 8πt e -n 2 τ /8t + 2 m≥1 e -(m+n) 2 τ /8t . (31)
It is useful to consider the Poisson-Jacobi inversion formula [START_REF] Whittaker | A Course of Modern Analysis[END_REF] ∞

m=-∞ e -πtm 2 -2πmta = t -1/2 e πa 2 t 1 + 2 ∞ m=1
e -πm 2 /t cos (2πma) [START_REF] Angstmann | Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains[END_REF] to rewrite equation ( 31) as an asymptotic expansion

E(n, t) ≃ 1 - τ 8πt 1 -e -n 2 τ /8t + 2 n m=1 e -m 2 τ /8t ≃ 1 -n τ 2πt . ( 33
)
From this long-time approximation it is easy to recover the same asymptotic behavior of the concentration in equation ( 29).

Let us now apply this technique to the critical value β = 1/2 with the same initial conditions. The generating function ( 26) is equal, after performing the summation over k, to

G(z, t) = F (wr(t)) (1 -w 2 )r(t) 1 -w 2 r 2 (t) exp 4 ατ w 2 r 2 (t) log (1 -w 2 )r 2 (t) 1 -w 2 r 2 (t) . ( 34
)
The probability E(n, t) is then equal to the contour integral

E(n, t) = dz 2iπz n+1 1 + 2 m≥1 1 + wr(t) 1 -wr(t) m (35) × (1 -w 2 )r(t) 1 -w 2 r 2 (t) exp 4 ατ w 2 r 2 (t) log (1 -w 2 )r 2 (t) 1 -w 2 r 2 (t) .
As before, for large times, we can use the change of variable z = e iθ and expand the previous integrand around θ = 0, assuming this is the main contribution to the integral. In term of the variable w, we have the approximation w ≃ -iθ/2 and therefore

E(n, t) ≃ ∞ -∞ dθ 2π e -inθ + 2 m≥1 e -i(mr(t)+n)θ × (1 + θ 2 /4)r(t) 1 + θ 2 r 2 (t)/4 exp - θ 2 r 2 (t) ατ log (1 + θ 2 /4)r 2 (t) 1 + θ 2 r 2 (t)/4 . (36)
In this expression we perform the variable change r(t

)θ → θ E(n, t) ≃ ∞ -∞ dθ 2π e -inθ/r(t) + 2 m≥1 e -i(m+n/r(t))θ × (1 + θ 2 /4r 2 (t)) 1 + θ 2 /4 exp - θ 2 ατ log r 2 (t) + θ 2 /4 1 + θ 2 /4 . ( 37
)
This integral can be further simplified if we keep the dominant contributions in 1/r(t) and discard the terms in 1/r 2 (t): We can use in the previous the asymptotic result

E(n, t) ≃ ∞ -∞ dθ 2π e -inθ/r(t) + 2 m≥1 e -i(m+n/r(t))θ exp - 2θ 2 ατ log r(t) (38)
m≥1 me -πǫm 2 ≃ ǫ -1 ∞ 0 dsse -πs 2 ≃ (2πǫ) -1 ( 39
)
when ǫ is small, to obtain the following expansion for the interval probability and concentration

E(n, t) ≃ 1 - n r(t) ατ 2π log r(t) , r(t)c(t) ≃ ατ 2π log r(t) ∝ log(t) -1/2 . ( 40
)
We recover in this case the asymptotic result in the continuous case given by equation [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF]. A more direct and simpler analysis of the discretized diffusion process is to start with a modified form of the transformed equation ( 10)

τ dE(n, u) du = 2∆E(n, u) (41)
the solution of which is given formally by equation [START_REF] Whittaker | A Course of Modern Analysis[END_REF], or

E(n, u) = e -4u/τ m≥1 E(m, 0) I n-m 4u τ -I n+m 4u τ +2 m≥1 I n+m 4u τ + I n 4u τ (42)
with E(m, 0) = E(m, 0). Then, after substituting the original time and space variables u → u(t) and n → nr -1 (t), we obtain

E(n, t) = e -4u(t)/τ m≥1 E(m, 0) I n/r(t)-m 4u(t) τ -I n/r(t)+m 4u(t) τ +2 m≥1 I n/r(t)+m 4u(t) τ + I n/r(t) 4u(t) τ . ( 43
)
From this result, the concentration is given, with a lattice initially filled with particles as initial condition, by the expression

c(t) = 1 -e -4u(t)/τ 2 m≥1 I 1/r(t)+m 4u(t) τ + I 1/r(t) 4u(t) τ . ( 44
)
The number of remaining particles is given by N p (t)/N 0 = r(t)c(t), with N 0 the number of initial lattice sites which is equal to the initial number of particles. The comparison with the continuous case, equation [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF], is displayed in the inset of figure 2 for β = 0.2.

We have set √ Dc 0 = a 2 /τ N 0 /(N 0 a) = 1/τ and expressed the other parameters with ατ = 1. As expected, the large time limit of both cases are equivalent.

Analogy with geometrical expanding structures

In this section we would like to discuss the applicability of the previous results to the case of growth models on particular geometrical structures in two dimensions. We can indeed interpret the time dilatation process as the increase of the available space a random particle is allowed to move into at every time step, due only to geometrical constraints. As an example, let us consider the properties of a directed growth structure on a disk domain, see references [START_REF] Ali | Scale-invariant growth processes in expanding space[END_REF][START_REF] Ali | Interacting particle systems in timedependent geometries[END_REF][START_REF] Vot | Encounter-controlled coalescence and annihilation on a one-dimensional growing domain[END_REF], such as depicted in figure 5(a). We can see that the trajectory tends to develop on a circle of growing radius and therefore in a larger space as time increases. The equation of diffusion is therefore modified by the expansion of the space. Following the same idea as in reference [START_REF] Ali | Scale-invariant growth processes in expanding space[END_REF], we note ρ k = ρ 0 +kdρ the radius of the circle where the particle or arm is located at step k with increments dρ. Originally the particle is located at radius ρ 0 , and we consider a Brownian motion such as equation ( 2), but with variable y k ∈ [0, 2πρ k ] measuring the displacement length of the particle on the corresponding circle

y k = r k r k-1 (y k-1 + ξ k ) . ( 45
)
The growth factor r k = (1 + β -1 kdρ/ρ 0 ) β is needed to take into account the expansion of the radius domain at every step. For a disk geometry [START_REF] Ali | Scale-invariant growth processes in expanding space[END_REF], such as considered in figure 5(a), β is unity, but here we generalize this model to any geometry characterized by the scaling exponent β, as in equation ( 1), characterizing the expansion rate. The mapping onto a linear or strip domain of fixed width L 0 = 2πρ 0 , see figure 5(b), is made possible by considering the transformed process

x k = (ρ 0 /ρ k )y k ∈ [0, L].
The new variables x k satisfy a modified Brownian motion

x k = x k-1 + ξ k r k-1 , x k = k i=1 ξ i r i-1 + x 0 . ( 46
)
This equation is similar to a classical free Brownian trajectory except that the noise is multiplied by a non-constant scaling factor which modifies the properties of the dynamics. However it is possible to map the radius ρ k into an effective height h k , see figure 5(b), with the condition h 0 = 0, by considering the variance of the x k , and using the same noise definition as in equation ( 4):

ξ i ξ j = 2a 2 δ i,j -3 -2 -1 0 1 2 3
h k = (x k -x 0 ) 2 = 2a 2 k i=1 1 r 2 i-1 .
(47)

In the continuous limit kdρ → ρρ 0 and a 2 /dρ → D we obtain

h(ρ) = 2D βρ 0 1 -2β 1 + β -1 (ρ -ρ 0 )/ρ 0 1-2β -1 , β = 1 2 h(ρ) = Dρ 0 log [(ρ + ρ 0 )/2ρ 0 ] , β = 1 2 (48)
which is equivalent to the effective or comoving diffusive length given by equation ( 5) with α = 1/ρ 0 . The role played by the time variable is replaced by the radius difference ρρ 0 of the growing structure and the mapping in figure 5 shows that the projected height h(ρ) takes a finite value when β > 1/2, in the limit ρ → ∞. The corresponding relation between the exponent γ in reference [START_REF] Ali | Interacting particle systems in timedependent geometries[END_REF] and β = (2γ) -1 is only valid for the growth structure on the disk, figure 5(a), and for a Brownian noise γ = 1/2 or β = 1.

In reference [START_REF] Ali | Interacting particle systems in timedependent geometries[END_REF], the authors consider instead the growth of structures on a disk with different kinds of noise (Levy flights, fractional Brownian motion, see also [START_REF] Øksendal | Stochastic Differential Equations: An Introduction with Applications[END_REF][START_REF] Hahn | Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion[END_REF]) with γ related to the fluctuation scaling δy k ∼ (δρ k ) γ of the arm structures of figure 5(a), and therefore the previous scaling relation between β and γ is not applicable in the general case. Instead, in the present study, β is a geometrical exponent related to the domain curvature while the noise is purely Gaussian.

For β = 1/2, equation ( 48) is the usual conformal mapping from the outside circle ρ ≥ ρ 0 to the strip of width L 0 and h(ρ) ≥ 0. Also, for ρ close to ρ 0 we have the local linear relation h(ρ) ≃ 2D(ρρ 0 ) between the two domains at small times. Finally, it is worth noticing that h k is the time change such that if

B k = k i=1 ξ i is a Wiener process, B h k = k i=1 ξ i /r i-1 is also a Wiener process B h k B h l = min(h k , h l )
with respect to the new variable h k .

Modelization in term of fractional diffusion

We have seen that a space dilatation with algebraic or geometrical exponent β < 1/2 leads to the one-particle absorbing state with a power law decay proportional to t β-1/2 representing a subdiffusive dynamics. We can try to modelize this abnormal diffusion using the formalism of fractional dynamics [START_REF] Yuste | Subdiffusion-limited A + A reactions[END_REF][START_REF] Ali | Interacting particle systems in timedependent geometries[END_REF][START_REF] Angstmann | Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains[END_REF]. This idea is to incorporate phenomenologically the effect of the expansion or dilution into a single fractional operator acting on the diffusion process itself, modifying the effective exponent of the diffusion. We would like to see in details how the diffusion length is characterized and if the fractional formalism describes accurately the previous results. The time evolution of the empty intervals, equation ( 9), can instead be described by considering a fractional diffusion equation for the IPDF [START_REF] Yuste | Subdiffusion-limited A + A reactions[END_REF]

∂E(x, t) ∂t = 0 D 1-2ν t 2K ν ∂ 2 E(x, t) ∂x 2 = 2K ν Γ(2ν) ∂ ∂t t 0 dt ′ (t -t ′ ) 1-2ν ∂ 2 E(x, t ′ ) ∂x 2 (49)
where the Grünwald-Letnikov operator 0 D 1-2ν t (see [START_REF] Hilfer | Applications of Fractional Calculus in Physics[END_REF] for the technical details) ensures that the mean squared displacement of a particle, or fractional diffusive length, is subdiffusive:

ℓ 2 f (t) ≃ 2K ν t 2ν /Γ(1+2ν) with 0 < ν ≤ 1/2.
We assume that equation (49) describes correctly the physics of the model for the IPDF given by equation [START_REF] Ali | Scale-invariant growth processes in expanding space[END_REF], and that ℓ f (t) is associated to ℓ(t), equation [START_REF] Russo | Onedimensional diffusion-limited relaxation of photoexcitations in suspensions of single-walled carbon nanotubes[END_REF]. Exponents ν and β are therefore related by the identity ν = 1/2β, and constant K ν can be identified here with the diffusion coefficient. We find in particular when t ≫ 1

K ν ≃ D(1 -2ν) 4αν 2α 1 -2ν 2ν Γ(1 + 2ν). ( 50
)
It is known that the solution of equation (49) gives the concentration expression in term of a Mittag-Leffler function [START_REF] Yuste | Subdiffusion-limited A + A reactions[END_REF] which has the following asymptotic expansion [START_REF] Mainardi | On some properties of the Mittag-Leffler function E α (-t α ), completely monotone for t > 0 with 0 < α < 1[END_REF] N p (t)

N 0 = E ν -c 0 2K ν t ν ≃ (2K ν ) -1/2 c 0 Γ(1 -ν) t -ν . (51)
In our model, equation [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF], we have instead asymptotically, in the regime β < 1/2

N p (t) N 0 ≃ Γ(1 + 2ν) 1/2 c 0 (2πK ν ) 1/2 t -ν . ( 52
)
The two asymptotic behaviors give the same time dependence, but differ by the two numerical coefficients Γ(1ν) -1 and [Γ(1 + 2ν)/π] 1/2 , depending on the equation considered to model the diffusion. These two coefficients coincide only for the normal Brownian diffusion (ν = 1/2 or β = 0 where K 1/2 = D). We should also notice that the particular case E 1/2 (-x) = e x 2 erfc(x) corresponds to the IPDF found in equation [START_REF] Hallatschek | Genetic drift at expanding frontiers promotes gene segregation[END_REF]. However equations ( 49) and (51) do not describe the localized regime for which β ≥ 1/2 (in this case ν would be negative). In the limit β = 1/2 or ν → 0, we can however recover the logarithmic behavior of the effective length as K ν ≃ D/(4αν) and ℓ 2 f (t) ≃ 2K ν e 2ν log(t) ≃ 4νK ν log(t) ≃ (D/α) log(t). Let us first discuss the subdiffusive regime which can be studied in the framework of CTRW models [START_REF] West | Fractional diffusion and Lévy stable processes[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF][START_REF] Barkai | From continuous time random walks to the fractional Fokker-Planck equation[END_REF][START_REF] Vot | Continuous-time random walks and Fokker-Planck equation in expanding media[END_REF] related to fractional diffusion, where time and space steps are distributed from a probability function. Let consider a random process where time steps, or waiting times τ i , are taken from the following distribution

ϕ(τ i ) = τ 2ν c Γ(2ν) e -τc/τ i τ 1+2ν i ( 53
)
where τ c is some short time cutoff that is necessary to normalize the distribution and ν > 0 an arbitrary exponent not necessary equal to 1/2β. The distribution has a finite mean τ = τ i , only if ν > 1/2: τ = τ c /(2ν -1). We should notice that a different distribution such as ϕ(τ i ) = 2ντ 2ν c /(τ c + τ i ) 1+2ν would give similar results. We also assume that space steps a i are taken from a Gaussian distribution λ(a i ) = e -a 2 i /4a 2 / √ 4πa 2 [START_REF] Vot | Continuous-time random walks and Fokker-Planck equation in expanding media[END_REF]. Its Fourier transform is given by λ(k) = e -k 2 a 2 ≃ 1a 2 k 2 . We consider in this context the Green function W (x, t), which is the probability to find a particle at location x at time t. For small momenta, the Fourier-Laplace transform W (k, s) of W (x, t) satisfies a Montroll-Weiss equation [START_REF] Montroll | Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries[END_REF][START_REF] Metzler | Levy statistics and anomalous transport: Levy flights and subdiffusion[END_REF][START_REF] Vot | Continuous-time random walks and Fokker-Planck equation in expanding media[END_REF] s W (k, s) = 1φ(s) + s φ(s) λ(k) W (k, s)

which can be rewritten in the present case as

s W (k, s) = 1 -a 2 k 2 s φ(s) 1 -φ(s) W (k, s) (55)
where

φ(s) = 2(τ c s) ν Γ(2ν) K 2ν (2 √ τ c s) . (56)
In the long time limit or for small s, we find the following expansion

φ(s) ≃ 1 - Γ(1 -2ν) 2νΓ(2ν) (τ c s) 2ν , ν < 1 2 , φ(s) ≃ 1 -(τ c s) log e 1-2γ τ c s , ν = 1/2, φ(s) ≃ 1 - τ c s 2ν -1 , ν > 1 2 . ( 57
)
Using the first term of the previous equation when ν < 1/2, we have

s W (k, s) ≃ 1 - a 2 k 2 τ 2ν c 2νΓ(2ν) Γ(1 -2ν) s 1-2ν W (k, s) (58)
from which we can inverse the Laplace and Fourier transforms, and obtain the fractional diffusion equation for the subdiffusive regime ν < 1/2

∂W (x, t) ∂t ≃ K ν Γ(2ν) ∂ ∂t t 0 dt ′ (t -t ′ ) 1-2ν ∂ 2 W (x, t ′ ) ∂x 2 , K ν = 2νa 2 Γ(2ν) τ 2ν c Γ(1 -2ν) . ( 59
)
We have identified a diffusive factor K ν , as in equation ( 49), but without the factor 2 which is normally found for the IPDF. For the marginal case ν = 1/2 we find

∂W (x, t) ∂t ≃ ∂ ∂t t 0 dt ′ 1 0 K ν du Γ(1 -u) e 1-2γ t -t ′ u ∂ 2 W (x, t ′ ) ∂x 2 , K ν = a 2 τ c . ( 60
)
This expression comes from the identity when s is small

L -1 s -1 log s = ∂ t 1 0 du t -u Γ(1 -u) ≃ L -1 - 1 log s (61)
which is correct in the long time limit. In the diffusive case ν > 1/2, we recover the usual equation of diffusion

∂W (x, t) ∂t = K ν ∂ 2 W (x, t) ∂x 2 , K ν ≃ a 2 (2ν -1) τ c = a 2 τ = D. (62)
The fractional diffusive length ℓ f (t) can be evaluated using the definition in term of the Laplace transform

L ℓ 2 f (t) = - ∂ 2 W (k, s) ∂k 2 k=0 = 2a 2 φ(s) s(1 -φ(s)) . (63)
Using the three expansions of equation (57), we obtain the following results in the long time limit after taking the inverse Laplace transforms

ℓ 2 f (t) ≃ 2a 2 τ 2ν c Γ(1 -2ν) t 2ν , ν < 1/2, ℓ 2 f (t) ≃ 2a 2 τ c t log (e 1-2γ t/τ c ) , ν = 1/2 ℓ 2 f (t) ≃ 2a 2 (2ν -1) τ c t = 2Dt, ν > 1/2. ( 64
)
In the marginal case, ν = 1/2, we have used the identities

L -1 1 -s -2 log s = 2 0 du t 1-u Γ(2 -u) ≃ L -1 - 1 s 2 log s , 2 0 du t 1-u Γ(2 -u) ≃ t log(t) , for t ≫ 1. (65)
The fractional diffusion dynamics characterized by equations (64) represents therefore a subdiffusive process for which ℓ 2 f (t) cannot exceed t, as in equation ( 5). When ν > 1/2, the time distribution for the τ i , equation (53), has a finite mean which implies that the diffusive process is Brownian. In the opposite case, when ν < 0, we can argue that the waiting times are infinite and therefore the particles appear to be localized. From the point of view of Lévy process, we can also estimate the fractional diffusive length using simple scaling relations. If N t is the number of the particle steps, then ℓ 2 f (t) ∼ a 2 N t . The time duration for performing N t independent steps scales like [START_REF] Valentin | Limit theorems of probability theory: sequences of independent random variables[END_REF] according to the distribution properties of equation ( 53) which has a divergent mean value when ν < 1/2, and proportional to N t when ν > 1/2. We therefore recover the fact that ℓ 2 f (t) ∼ t 2ν and ℓ 2 f (t) ∼ t respectively. The fractional case describing subdiffusive processes can be compared to the effective diffusive length ℓ(t) of the expansion model obtained in equation ( 5). Asymptotically we deduce that

t = τ 1 + • • • + τ Nt ∼ N 1/2ν t
ℓ 2 (t) ∼ t 1-2β , β < 1/2, ℓ 2 (t) ∼ log(t), β = 1/2, ℓ 2 (t) ∼ 1, β > 1/2. ( 66
)
Only the first case β < 1/2 could be identified to the fractional diffusion, equation (64), when ν < 1/2 by substituting ν = 1/2-β. However, it is more interesting to consider the case where ν = β in equation ( 64). Then the product ℓ 2 (t)ℓ 2 f (t) is always proportional to t for any value of β, i.e. proportional to the squared of the classical Brownian diffusive length. This reflects the duality between the diffusion on an expanding space and the fractional model of waiting times distributed with an algebraic law.

If we consider instead the physical length ℓ phys (t) = r(t)ℓ(t) rather than the comoving length ℓ(t), we have the following long-time behavior

ℓ 2 phys (t) ∼ t, β < 1/2, ℓ 2 phys (t) ∼ t log(t), β = 1/2, ℓ 2 phys (t) ∼ t 2β , β > 1/2. ( 67
)
It is clear that equation (67) represents a superdiffusive process, in opposition to equation (64), and with the physical length ℓ phys (t) scaling like the expansion rate r(t) when β > 1/2. Instead of considering a distribution of waiting times, let assume that the time steps are constant and equal to τ , and that the space increments a i s are taken from the distribution λ(a i ) whose Fourier transform is equal to

λ(a i ) = (1 + ν) sin π 2(1+ν) πa 1 1 + |a i /a| 2+2ν (68)
where ν > 0 is the exponent of the algebraic decay and a a typical length. For ν ≤ 1/2 the distribution has an infinite variance. Its Fourier transform is given by λ

(k) = 2(1 + ν) sin π 2(1+ν) π ∞ 0 du cos(kau) 1 + u 2+2ν . ( 69
)
The behavior of λ(k) for small momenta depends on the value of ν. We can rewrite the previous expression in the case ν > 1/2 as λ(k) = 2(1 + ν) sin (71)

The last integral converges around 0 and at infinity when ν < 1/2. Using φ(s) = e -τ s ≃ 1τ s we can solve the Montroll-Weiss equation (54) and find W (k, s) ≃ 1 s + τ -1 |a ν k| 1+2ν , ν < 1/2.

(72)

The Laplace inverse gives simply

L -1 ( W (k, s)) = exp(-τ -1 |a ν k| 2ν+1 t) (73)
and the Green function is expressed as

W (x, t) = τ t 1/(2ν+1)
du 2πa ν e -iu(τ /t) 1/(2ν+1) x/aν -|u| 2ν+1 = a -1 ν τ t

1/(2ν+1)

F ν τ t 1/(2ν+1)
x/a ν .

The function F ν is a Levy distribution [START_REF] Metzler | Levy statistics and anomalous transport: Levy flights and subdiffusion[END_REF] with exponent 2ν + 1. When ν = 1/2 we recover the Gaussian density F 1/2 (y) = e -y 2 /4 /2 √ π and the fact that the diffusive length scales like t 1/2 . For ν < 1/2 the variance of the function λ is infinite and the diffusive length is not defined. However the scaling of the function (74) would formally gives a length scale growing like t 1/(2ν+1) , with an exponent larger than 1/2 and less than unity. This superdiffusive scaling can be achieved using a truncated Levy flight process by discarding the large deviations beyond a cutoff length [START_REF] Mantegna | Stochastic Process with Ultraslow Convergence to a Gaussian: The Truncated Lévy Flight[END_REF]. Although the Gaussian behavior should be recovered in the large time limit, we expect to see a superdiffusive regime for some time interval. The diffusive length squared should indeed scale like Nt i=1 a 2 i ∼ N 2/(2ν+1) t ∼ t 2/(2ν+1) for N t time steps of duration τ .

Conclusion

We have investigated in this paper the limiting effect of an inflating space on the reaction process of coagulation-diffusion for Brownian particles, in particular we have obtained exact solutions when the space is discrete for a general initial distribution of particles and for a given expansion law characterized by the algebraic exponent β, using generating functions. The results of the continuous model are recovered in the large time limit. This model possesses two regimes which can be understood in the steady state limit as a second order phase transition between an absorbing state N p = 0 and a localized state N p = 0, with a slowing down phenomenon at the critical point β = 1/2, β being identified with an inverse temperature.

An analogy is made with growth systems constrained by radial geometries and for which the noise is Gaussian. We extended the problem to expanding geometries which generalizes the radial case.

We finally compare these results with fractional models of waiting time distributions, and found a duality relation between the effective diffusive lengths. More precisely, for the same class of exponent, the product of the effective length ℓ(t) in the model of space expansion with the diffusive length ℓ f (t) given by the fractional representation of the diffusion operator is always proportional to t 1/2 , which is the effective length of the pure Brownian motion, even at criticality.

Figure 1 .

 1 Figure 1. Numerical evaluation of the ratio of remaining number of particles N p /N 0 for several values of β. Initial parameters are N 0 = L 0 = 1000 and α = 10 -4 . The dashed line represents the asymptotic power law t -ν = t -1/2 when β = 0. Inset: Decay exponent ν extracted from the asymptotic behavior as function of β. Red line is the expected theoretical result: ν = 1/2β.

Figure 2 .

 2 Figure 2. Analytical evaluation of the remaining number of particles from equation (18), for several values of β. Parameters are chosen such that D = 1/(8N 0 ), c 0 = 1 and α = 10 -4 in dimensionless units. Inset: Comparison between expression (18) (green line) and the discrete solution (dashed line), see equation (44), for β = 0.2 and ατ = 1.

Figure 3 .

 3 Figure 3. Numerical evaluation of the ratio N p /N 0 in the exponential case, for several expansion rates α. Initial parameters are N 0 = L 0 = 1000. Inset: Number of remaining particles as function of λ. The red curve is the fit using equation (19) with λ = (8N 0 α) -1/2 .

Figure 4 .

 4 Figure 4. Evaluation of the analytical ratio of the remaining particles N p /N 0 in the exponential case, equation (19), for several parameters λ = c 0 D/α, from λ = 2 (top curve) to λ = 10 (bottom curve), and α = 10 -4 . The dashed red curve is the comparison with the diffusive case only, see equation (20). Inset: Number of remaining particles as function of λ.

Figure 5 .

 5 Figure 5. Brownian motion on a circle (a) and mapping onto a linear domain (b), see references [9, 29] figure 1. ρ k is the radius of the arm structure at time step k, and the radius maps onto an effective height h k in the linear domain of fixed width L 0 = 2πρ 0 .

1 + 2 0 ( 1 -

 1201 u 2+2ν du ≃ 1 -(a ν k) 2 , ν > 1/2 (70) with a 2 ν = (4 cos 2 [π/(2 + 2ν)] -1)a 2 /2.The diffusion is Brownian with a length scale proportional to t 1/2 . In the case ν < 1/2, the integral in the second line of equation (70) is divergent. We can write insteadλ(k) ≃ 1 + 2(1 + ν) sin π cos(u))du u 2+2ν = 1 -(a ν k) 1+2ν , ν < 1/2.

Acknowledgments

This work was supported by Brain Pool Program through the National Research Foundation of Korea (Grant No. NRF-2018H1D3A2065321) and in part through the Basic Science Research Program (Grants No. 2016R1D1A1A09917318 and 2019R1F1A1046285).


Nicolas Rougerie
email: nicolas.rougerie@grenoble.cnrs.fr

ON TWO PROPERTIES OF THE FISHER INFORMATION

published or not. The documents may come

The Fisher information of a symmetric probability measure of N variables (see Definition 2.1 below) is known to be:

A. Superadditive: the information of the full measure is not smaller than the sum of the informations of marginals with n and Nn variables.

B. Affine linear in the large N limit: for probability measures that have a limit N → ∞, the natural limiting information (mean, or level-3, information) is an affine functional.

A has been proved first in [START_REF] Carlen | Superadditivity of Fisher's information and logarithmic Sobolev inequalities[END_REF]Theorem 3] and then, by another method, in [8, Lemma 3.7]. B is deduced in [12, Proposition 3] from the corresponding property for the mean entropy, originating in [START_REF] Robinson | Mean entropy of states in classical statistical mechanics[END_REF]. In [START_REF] Hauray | On Kac's chaos and related problems[END_REF]Section 5.3] a different proof is provided, based on a general abstract linearity lemma. The analogues for some fractional variants of Fisher's information are obtained in [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF][START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF].

This note provides alternative proofs of these properties. If the Fisher information of a probability density µ N ∈ P sym (R dN ) is interpreted as a quantum kinetic energy for the quantum state |Ψ N Ψ N |, orthogonal projector on the "quantum wave-function" As for the motivations behind proving A and B, they mostly come from the study of mean-field limits of large systems of statistical mechanics, classical and quantum. Indeed, they do not seem to be of use in the theory of sufficient statistics, where the Fisher information originates. We however note that any quantity interpreted as an "information" should certainly satisfy A. B has been known to hold for the entropy for a long time [START_REF] Robinson | Mean entropy of states in classical statistical mechanics[END_REF], and it seems natural to ask its equivalent for the Fisher information.

Ψ N = √ µ N ∈ L 2 sym (R dN ),
Both properties are crucial to Kiessling's approach [START_REF] Kiessling | The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion[END_REF] of the mean-field limit of bosonic ground states (see [19, 20, Appendix A] for review), which involves interpreting the quantum kinetic energy as a classical Fisher information. In fact we exploit here the reverse of Kiessling's point of view.

In [START_REF] Fournier | Propagation of chaos for the 2d viscous vortex model[END_REF], Fisher information bounds are used to control the mean-field limit of a classical statistical mechanics system with stochastic diffusions (see [START_REF] Hauray | Limite de champ moyen et propagation du chaos pour des systmes de particules, limites gyro-cintique et[END_REF] for review). The method is general and has been adapted to other models, e.g. in [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF][START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF]. Briefly, the entropy production1 along the flow is controlled first. The Fisher information is naturally linked to variations of the entropy2 , and one then deduces a control of the former quantity. This has several important applications, one of which relies on A and B above. Briefly, A allows to pass to the large N limit and control the mean information of limiting objects. Then B implies that the de Finetti-Hewitt-Savage mixing measure of the limit is concentrated on probability measures with finite Fisher information. Uniqueness theorems for the mean-field equation in the latter class can then be put to good use.

Acknowledgments. Thanks to Samir Salem, conversations with whom motivated the write-up of this note, and to Mathieu Lewin for discussions on related topics a while ago. Financial support was provided by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement COR-FRONMAT No 758620).

Definitions and Results

We are concerned with classical mechanics states, symmetric probability measures µ N ∈ P sym (R dN ) for the distribution of N indistinguishable particles living say in R d . We will freely identify measures and their densities with respect to Lebesgue measure on R dN . Symmetric here means

µ N (x 1 , . . . , x N ) = µ N (x σ(1) , . . . , x σ(N ) ) (2.1)
for almost all X N = (x 1 , . . . , x N ) ∈ R dN and any permutation σ of the N indices. In the applications we have in mind, X N can be a collection of spatial coordinates (this is the case for bosonic mean-field limits as considered in [START_REF] Kiessling | The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion[END_REF]) or a collection of velocity variables (this is the case for the applications in kinetic theory [START_REF] Hauray | On Kac's chaos and related problems[END_REF][START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF][START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF]).

The quantities of our interest are as given in the Definition 2.1 (Fisher informations).

For µ N ∈ P sym (R dN ) we define

• the Fisher information

I 1 [µ N ] = √ µ N N j=1 -∆ x j √ µ N L 2 (R dN )
.

(2.2)

• the fractional Fisher information of order 0 < s < 1

I s [µ N ] = √ µ N N j=1 -∆ x j s √ µ N L 2 (R dN )
.

(2.3)

Remarks.

1. The following equivalent definitions are well-known. For the usual Fisher information (the first one is maybe the most commonly used) we have

I 1 [µ N ] = 1 4 R dN |∇ log µ N | 2 µ N = 1 4 R dN |∇µ N | 2 µ N = R dN |∇ √ µ N | 2 = N j=1 R dN |k j | 2 √ µ N (k) 2 dk (2.4)
and for the fractional Fisher information:

I s [µ N ] = N j=1 R dN |k j | 2s √ µ N (k) 2 dk = C d,s N R d(N+1) µ N (x, x 1 , . . . , x N -1 ) -µ N (y, x 1 . . . , x N -1 ) 2 |x -y| d+s dxdydx 1 . . . dx N -1 .
(2.5)

Here hat-bearing functions stand for Fourier transforms and C d,s is a constant only depending on d and s. That these various definitions are equivalent either follows from straightforward calculations or is proved in standard textbooks, such as [15]. The precise value of the constant C d,s is of no concern to this note but can be found in [15,[START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF][START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF].

2.

It follows from results of [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Masja | Über äquivalente normierung der anisotropen Funktionalraüme H µ (R n )[END_REF][START_REF]Limiting embedding theorems for W s,p when s ↑ 1 and applications[END_REF][START_REF] Maz | On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces[END_REF] that

I 1 [µ N ] = C d lim s↑1 (1 -s)C -1 d,s I s [µ N ] (2.6)
with

C d = S d-1 cos θ dσ -1
.

Here S d-1 is the euclidean sphere equipped with its Lebesgue measure dσ and θ = θ(σ) represents the angle of σ with respect to the vertical axis. This implies that I 1 ≡ I is a natural limit case of I s for s → 1.

3.

Other types of fractional Fisher informations are discussed in [START_REF] Toscani | The fractional Fisher information and the central limit theorem for stable laws[END_REF][START_REF]Score functions, generalized relative Fisher information and applications[END_REF][START_REF]The information-theoretic meaning of Gagliardo-Nirenberg type inequalities[END_REF], in connection with statistics, and [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF], in connection with the fractional heat flow.

4.

In [START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF] the fractional Fisher information is defined with an extra "cut-off"

I s,γ [µ N ] = N j=1 √ µ N |χ(x j )(-∆ x j ) s χ(x j )| √ µ N L 2 with χ(x) = (1 + |x| 2 ) 2γ
, γ < 0. This does not significantly change the structure of the object, nor the proofs of the results below. Indeed the "kinetic energy" u|χ(-∆) s χ|u still enjoys the properties we need (see Properties 3.1), of which convexity as a function of |u| 2 and affinity as a function of the orthogonal projector |u u| are the most crucial. We leave the adaptations to the reader.

Define now, for any integer n N , the n-th marginal/reduced density of a measure µ N ∈ P sym (R dN ) as the probability measure on R dn with density

µ (n) N (x 1 , . . . , x n ) := R d(N-n) µ N (x 1 , . . . , x n , y n+1 , . . . , y N )dy n+1 . . . dy N .
(

The first result we provide an alternative proof for is the superadditivity of the functionals from Definition 2.1:

Theorem 2.

(Superadditivity of Fisher informations).

Let n < N be two integers and µ N ∈ P sym (R dN ). We have that

I s [µ N ] I s µ (n) N + I s µ (N -n) N (2.8)
for any 0 < s 1.

Remarks. 1. For s = 1, Carlen proved this via a Minkowski-like inequality. Hauray-Mischler use a dual formulation of the Fisher information (not mentioned in the remark after Definition 2.1) to recover the result. For s < 1 the result is obtained in [START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF] using the last formulation in (2.5). As per (2.6), this also implies the result for s = 1.

2.

The short proof we provide uses standard tools of quantum mechanics: reduced density matrices and convexity of µ

N → I s [µ N ]. 3. A useful consequence is that, if N is an integer times n 1 N I s [µ N ] 1 n I s [µ (n) N ]. (2.9)
In statistical mechanics one is often interested in the limit of large particle numbers, N → ∞ , maybe with other parameters of the model scaled appropriately. Then our classical states turn into symmetric probability measures over infinite sequences, µ ∈ P sym (R dN ). One can also be interested in the limits N → ∞ with fixed n of the marginals (2.7). It can then be useful to have a notion of "mean Fisher information" (sometimes also refered to as level-3 information): Definition 2.3 (Mean Fisher information). Let µ ∈ P sym (R dN ) be a symmetric probability measure over sequences in R d . Equivalently3 , let (µ (n) ) n be a sequence of symmetric probability measures over R dn satisfying the consistency condition

µ (n+1) (n) = µ (n) .
For 0 < s 1 the mean (fractional) Fisher information of µ is

I s [µ] := lim sup n→∞ 1 n I s [µ (n) ].
(2.10)

The existence of the lim sup follows from (2.9). It is in fact possible to see that the lim sup is both a sup and a lim, using Theorem 2.4.

The definition is in complete analogy with that of the mean entropy, originating in [START_REF] Robinson | Mean entropy of states in classical statistical mechanics[END_REF]. Perhaps surprisingly, this functional is affine, just as the mean entropy. We shall give an alternative proof of the Theorem 2.4 (The mean Fisher information is affine). Let µ ∈ P sym (R dN ) and P ∈ P(P(R d )) be its unique de Finetti-Hewitt-Savage measure, i.e., for all n 0

µ (n) = P(R d ) ρ ⊗n dP (ρ).
(2.11)

Assume that there exists a locally bounded V :

R d → R with V (x) -→ |x|→∞ +∞ such that R d V µ (1) < ∞.
(2.12)

Then, the mean Fisher informations from Definition 2.3 satisfy, for all 0 < s 1

I s [µ] = P(R d ) I s [ρ]dP (ρ) (2.13)
with I s [ρ] as in Definition 2.1 with N = 1.

Remarks.

1. In (2.12) I demand a bit more "confinement" than in previous versions of the statement [START_REF] Kiessling | The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion[END_REF][START_REF] Hauray | On Kac's chaos and related problems[END_REF][START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF]. This is harmless in applications, for the theorem is meant to be applied to limits N → ∞ of N -body classical states µ N . In order for a state with infinitely many particles to exist in the limit, a tightness argument of the type

R d V µ (1)
N < ∞, independently of N is usually needed.

2. Kiessling [START_REF] Kiessling | The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion[END_REF] proved the s = 1 case, using the better known [START_REF] Robinson | Mean entropy of states in classical statistical mechanics[END_REF] affinity of the mean entropy

S[µ] := -lim sup n→∞ 1 n R dn µ (n) log µ (n) .
The Fisher information is the derivative of the entropy along the heat flow. Since the latter is linear (as is differentiation), the affinity of the Fisher information follows.

3. Hauray and Mischler [START_REF] Hauray | On Kac's chaos and related problems[END_REF] gave another proof for s = 1. A first sanity check is to convince oneself that

I 1 2 ρ ⊗∞ 1 + 1 2 ρ ⊗∞ 2 = 1 2 I ρ ⊗∞ 1 + 1 2 I ρ ⊗∞ 2 (2.14)
where ρ ∞ is the measure over R dN with n-th marginal ρ ⊗n for any n. The reason for this is that, if

ρ 1 = ρ 2 , ρ ⊗N 1
becomes more and more alien ("orthogonal") to ρ ⊗N 2 for large N . Very briefly, the proof of [START_REF] Hauray | On Kac's chaos and related problems[END_REF] checks a more elaborate version of the "partial affinity" (2.14), and then applies a general abstract lemma implying full affinity.

4.

The same strategy is applied to the s < 1 case in [START_REF]Propagation of chaos for the boltzmann equation with moderately soft potentials[END_REF]. The abstract lemma applies mutatis mutandis, but the argument giving the partial affinity is different. The sanity check (2.14) can also be found in [19, 20, Appendix A]. [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF] that the variant mean fractional information 4 based on

Salem proves in

I s [µ N ] = N R d(N+1) Φ µ N (x, x 1 , . . . , x N -1 ), µ N (y, x 1 . . . , x N -1 )
|x -y| d+s dxdydx 1 . . . dx N -1 with Φ(x, y) = (xy)(log x -log y) enjoys similar properties as that we defined, in particular affinity. See [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF]Remark 3.4] for more comments on the relation between I s and I s and their respective uses. In particular, for the applications of [START_REF] Salem | Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases[END_REF], results on I s could serve as alternatives to those on I s .

Proofs

3.1. Preliminaries. Our point of view in this note is to think quantum mechanically, that is, in terms of L 2 functions and operators acting on them, rather than in terms of probability measures. Pick µ N ∈ P sym (R dN ). For it to have a finite Fisher information it must actually be a function. Define then

Ψ N = √ µ N , Γ N = |Ψ N Ψ N | (3.1)
The first object above is a bosonic wave-function, namely

Ψ N ∈ L 2 sym (R dN , C) satisfies Ψ N (x 1 , . . . , x N ) = Ψ N (x σ(1) , . . . , x σ(N ) )
in analogy with (2.1). The second object in (3.1) is the L 2 -orthogonal projector on the complex linear span of Ψ N . It is a bosonic state, i.e. a positive trace-class operator with trace 1, acting on L 2 sym (R dN , C). Note that we do not use the usual quantization of classical mechanics: we simply use that a classical state of position or velocity variables can be directly embedded in a quantum formalism.

We shall write Fisher informations as quantum kinetic energies of

Ψ N or Γ N I s [µ N ] = Ψ N |H N |Ψ N L 2 = Tr (H N Γ N ) (3.2)
with

H N = N j=1 h x j , h = (-∆) s
4 This is the derivative of the entropy along the fractional heat flow.

and h x j acting on the variable x j . We do not emphasize the dependence on s, for our proofs shall be based solely on the following Properties 3.1 (Quantum kinetic energies).

The kinetic energy

L 2 (R d , C) ∋ u → u|h|u ∈ R + ∪ {+∞}
with h as above is

(1) Positivity preserving

u|h|u |u| |h||u|
(2) Convex as a function of |u| 2 :

L 1 (R d , R + ) ∋ ρ → √ ρ|h| √ ρ ∈ R + ∪ {+∞} is convex. (3) With locally compact resolvent. For a locally bounded V : R d → R with V (x) -→ |x|→∞ +∞
the operator h+V has compact resolvent, i.e. (h+V +c) -1 is compact as an operator on L 2 , where c is a constant sufficiently large for the inverse to make sense.

Remarks. The first property for s = 1 is just the straightforward (at least for smooth functions u = |u|e iϕ ) identity

|∇u| 2 = |∇|u|| 2 + |u| 2 |∇ϕ| 2 ,
a particular case of the diamagnetic inequality [15,Theorem 7.12]. For s > 1 it follows immediately from the last definition in (2.5), the triangle inequality, and (2.6). Note that "positivity preserving" usually means something stronger (but also true in the case at hand), namely that the heat flow associated with h preserves positivity of functions. The property we require usually goes hand-in-hand with this "true" positivity-preserving property.

The second property can be found in [15,Theorems 7.8 and 7.13]. The third property follows from the Sobolev compact embedding in L 2 .

We also recall the notion of reduced density matrix, extending that of marginal (2.7). We define the n-th reduced density matrix Γ

(n) N by a partial trace Γ (n) N := Tr n+1→N Γ N . (3.3)
This is the operator on L 2 sym (R dn ) defined by the relation

Tr A n Γ (n) N = Tr A n ⊗ 1 ⊗(N -n) Γ N
for any bounded operator A n on L 2 sym (R dn ). Note that Γ N , as a trace-class (in particular, Hilbert-Schmidt) operator on L 2 has an integral kernel Γ N (x 1 , . . . , x N ; y 1 , . . . , y N ) = Ψ N (y 1 , . . . , y N )Ψ N (x 1 , . . . , x N ).

The integral kernel of Γ (n)

N is then given, formally, as

Γ (n) N (x 1 , . . . , x n ; y 1 , . . . , y n ) = R d(N-n) Γ N (x 1 , . . . , x n , z n+1 , . . . , z N ; y 1 , . . . , y n , z n+1 , . . . , z N )dz n+1 . . . dz N .
One use of these objects is that, if N is a multiple of n,

I s [µ N ] = Tr (H N Γ N ) = N n Tr H n Γ (n) N (3.4)
taking partial traces and using that H N is a sum of terms acting on one variable at a time.

3.2. Superadditivity. We now prove Theorem 2.2. The following considerations are very much in the spirit of the Hoffmann-Ostenhof 2 inequality [START_REF] Hoffmann-Ostenhof | Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules[END_REF][START_REF] Lewin | Mean-Field limit of Bose systems: rigorous results[END_REF]. From (3.2) and (3.3) we have, similarly to in (3.4),

I s [µ N ] = Tr H n Γ (n) N + Tr H N -n Γ (N -n) N (3.5)
where µ N and Γ N are related by (3.1). Now, Γ

N being a positive trace-class operator with unit trace, the spectral theorem implies the existence of an orthonormal basis (u j ) of

L 2 (R dn ) such that Γ (n) N = j λ j |u j u j |
where the positive numbers λ j add to 1. Thus

Tr H n Γ (n) N = j λ j u j |H n |u j
and, using Items 1 and 2 in Properties 3.1 we get

Tr H n Γ (n) N j λ j |u j ||H n ||u j | √ ρ n |H n | √ ρ n with ρ n := j λ j |u j | 2 .
The proof is concluded by observing that

ρ n = µ (n) N (3.6) with µ (n)
N the n-th marginal of µ N and arguing similarly for the second term of (3.5).

To see the truth of (3.6), identify any bounded function V n of n variables in R d with the corresponding multiplication operator on L 2 (R dn ). Then

R dn V n ρ n = Tr V n Γ (n) N = Tr V n ⊗ 1 ⊗(N -n) Γ N = Ψ N |V n ⊗ 1 ⊗(N -n) |Ψ N = R dN V n (x 1 , . . . , x n )|Ψ N (x 1 , . . . , x N )| 2 dx 1 , . . . dx N = R dn V n µ (n) N .
This completes the proof.

3.3. Affinity. Our proof of Theorem 2.4 is based on the manifest affinity of (3.2) as a function of Γ N , and the quantum de Finetti theorem, a generalization of the classical de Finetti-Hewitt-Savage theorem, see [START_REF] De | limites de champ moyen et condensation de Bose-Einstein, Les cours Peccot[END_REF][START_REF] Rougerie | De Finetti theorems, mean-field limits and Bose-Einstein condensation[END_REF] for review. Readers familiar with the classical theorem could note that the version of the quantum theorem I use has a proof (see [START_REF] Hudson | Locally normal symmetric states and an analogue of de Finetti's theorem[END_REF] and [14, Appendix A]) essentially identical to that of Hewitt-Savage [START_REF] Hewitt | Symmetric measures on Cartesian products[END_REF].

First observe that, as per Item 2 in Properties 3.1, Definition 2.3 and (2.11) we immediately have that

I s [µ] P(R d ) I s [ρ]dP (ρ).
We aim at a corresponding lower bound.

Define, for any N ∈ N, a bosonic quantum wave-function and a bosonic quantum state as

Ψ N = µ (N ) , Γ N = |Ψ N Ψ N |
where µ (N ) is the N -th marginal of µ ∈ P sym (R dN ). Let then, for k N , Γ

N ⇀ ⋆ γ (k) (3.7)
in the trace-class, as N → ∞. The latter being [START_REF] Simon | Trace ideals and their applications[END_REF][START_REF] Schatten | Norm Ideals of Completely Continuous Operators[END_REF] the dual of the compact operators (equiped with the operator norm, and where the duality bracket between two operators A, B is given by Tr(AB)), this means

Tr K k Γ (k) N → Tr K k γ (k) (3.8)
for any compact operator K k on L 2 sym (R dk ). By a diagonal extraction argument we can assume that, for all k 0 the sequences (Γ

N ) N converges weakly-⋆, along a common subsequence in N .

Let V be the potential such that (2.12) holds and

h V = h + V. Then Tr   k j=1 h V x j Γ (k) N   = k N µ (N ) N j=1 h x j + V (x j ) µ (N ) L 2 k I s [µ] + R d µ (1) V < ∞
by assumption. Hence, for all k 0 and any constant c > 0

Tr   k j=1 (h + V + c) x j Γ (k) N   C k,c
independently of N . Using cyclity of the trace, the positive operator L

1/2 k Γ (k) N L 1/2 k with L k = k j=1 (h + V + c)
x j is bounded in the trace-class. As above this implies that (modulo a futher extraction of a subsequence) it converges weakly-⋆ (in the sense of (3.8))

L 1/2 k Γ (k) N L 1/2 k ⇀ ⋆ L 1/2 k γ (k) L 1/2 k (3.9)
where the limit is identified by testing the convergence with smooth finite-rank operators and recalling (3.7). As per Item 3 in Properties 3.1, h + V + c has compact inverse provided c is chosen large enough. Consequently, so does L k . Then, by (3.9), Tr Γ

(k) N = Tr L -1 k L 1/2 k Γ (k) N L 1/2 k → Tr L -1 k L 1/2 k γ (k) L 1/2 k = Tr γ (k) .
This proves convergence of the trace-class norm

1 = Tr Γ (k) N → Tr γ (k)
and hence that actually

Γ (k) N → γ (k) (3.10)
strongly in the trace-class norm (see [START_REF] Dell'antonio | On the limits of sequences of normal states[END_REF] or [START_REF] Simon | Trace ideals and their applications[END_REF]Addendum H]). We now use the strong 5 quantum de Finetti theorem (see [START_REF] Hudson | Locally normal symmetric states and an analogue of de Finetti's theorem[END_REF][START_REF] Lewin | Derivation of Hartree's theory for generic mean-field Bose systems[END_REF] and [START_REF] Rougerie | De Finetti theorems, mean-field limits and Bose-Einstein condensation[END_REF][START_REF] De | limites de champ moyen et condensation de Bose-Einstein, Les cours Peccot[END_REF] for review) to obtain the existence and uniqueness of a probability measure Q over the unit sphere of Tr hγ (1) 5 "Strong" refers to the fact that we use strong trace-class convergence of reduced density matrices. This is the quantum analogue of the Hewitt-Savage theorem where one uses tightness of the marginals. There is also a "weak" quantum de Finetti theorem, relying only on weak-⋆ convergence of reduced density matrices. We do not use it here. so that, combining the two observations and (3.11),

L 2 (R d ) such that γ (k) = |u ⊗k u ⊗k |dQ(u). ( 3
I s [µ]
u|h|u dQ(u).

Recalling Item 1 of Properties 3.1 this yields

I s [µ] |u||h||u| dQ(u) = |u| 2 h |u| 2 dQ(u)
and the proof will be complete once we have proven the next lemma: This being true for all k and ϕ k , the proofs of the lemma and the theorem (approximating ρ → √ ρ h √ ρ by a sequence of continuous functions) are both complete.

1 .

 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2. Superadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3. Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1. Introduction

  both properties become quite natural. Roughly speaking A is a consequence of the convexity of the kinetic energy as a function of |Ψ N | 2 and B follows from its affinity as a function of |Ψ N Ψ N |. Date: August, 2020.

1

 1

N

  be the reduced density matrix (3.3) of Γ N . At fixed k, the sequence (Γ (k) N ) N is by definition bounded in the trace-class. Modulo a (not-relabeled) subsequence we thus have Γ (k)

Lemma 3 . 2 (→

 32 Identification of de Finetti measures). Let P ∈ P(P(R d )) be the classical de Finetti measure defined in Theorem 2.4 and Q ∈ P(L 2 (R d )) the quantum de Finetti measure defined in(3.11). We have, for any bounded continuous functionΦ over P(R d ) Φ(ρ)dP (ρ) = Φ(|u| 2 )dQ(u).Proof. The monomial functions Φ of the form (see[START_REF] Golse | On the Dynamics of Large Particle Systems in the Mean Field Limit[END_REF] Section 1.7])Φ(ρ) = M k,ϕ k (ρ) = R dk ϕ k ρ ⊗kwith k ∈ N and ϕ k bounded continuous over R dk generate a subalgebra of the continuous functions on the space of probability measures. This subalgebra is dense by the Stone-Weierstrass theorem and it thus suffices to test the claim against all the above monomials.We identify the function ϕ k with the multiplication operator thereby to writeM k,ϕ k (ρ)dP (ρ) = R dk ϕ k µ (k) = R dN ϕ k (x 1 , . . . , x k )µ (N ) (x 1 , . . . , x N )dx 1 . . . dx N = Tr ϕ k ⊗ 1 ⊗(N -k) Γ N = Tr ϕ k Γ (k) N Tr ϕ k γ (k) using Equations (2.11)-(2.7)-(3.1)-(3.3) and, in the last step, the fact that multiplication by a bounded function is a bounded operator to pass to the limit using (3.10). Since the left-hand side actually does not depend on N we deduceM k,ϕ k (ρ)dP (ρ) = Tr ϕ k γ (k) .As per(3.11) this implies the desired M k,ϕ k (ρ)dP (ρ) = M k,ϕ k (|u| 2 )dQ(u).

I follow the physicists' convention that the entropy is produced, not dissipated.

The Fisher information is the derivative of the entropy along the heat flow.

By a theorem of Kolmogorov.