Thierry Bodineau

Isabelle Gallagher

Laure Saint-Raymond

THE BROWNIAN MOTION AS THE LIMIT OF A DETERMINISTIC SYSTEM OF HARD-SPHERES

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1. Introduction 1.1. From microscopic to macroscopic models. We are interested here in describing the macroscopic behavior of a gas consisting of N interacting particles of mass m in a domain D of R d , with positions and velocities (x i , v i ) 1≤i≤N ∈ (D × R d ) N , the dynamics of which is given by (1.1)

dx i dt = v i , m dv i dt = - 1 ε j =i ∇Φ x i -x j ε ,
for some compactly supported potential Φ, meaning that the scale for the microscopic interactions is typically ε. We shall actually mainly be interested in the case when the interactions are pointwise (hard-sphere interactions): the presentation of that model is postponed to Section 2, see (2.1),(2.2).

In the limit when N → ∞, ε → 0 with N ε d = O(1), it is expected that the distribution of particles averages out to a local equilibrium. The microscopic fluxes in the conservations of empirical density, momentum and energy should therefore converge to some macroscopic fluxes, and we should end up with a macroscopic system of equations (depending on the observation time and length scales). However the complexity of the problem is such that there is no complete derivation of any fluid model starting from the full deterministic Hamiltonian dynamics, regardless of the regime (we refer to [START_REF] Olla | Hydrodynamical limit for a Hamiltonian system with weak noise[END_REF][START_REF] Esposito | Navier-Stokes equations for stochastic particle systems on the lattice[END_REF][START_REF] Quastel | Lattice gases, large deviations, and the incompressible Navier-Stokes equations[END_REF] for partial results obtained by adding a small noise in the microscopic dynamics).

For rarefied gases, i.e. under the assumption that there is asymptotically no excluded volume N ε d 1, Boltzmann introduced an intermediate level of description, referred to as kinetic theory, in which the state of the gas is described by the statistical distribution f of the position and velocity of a typical particle. In the Boltzmann-Grad scaling α ≡ N ε d-1 = O(1), we indeed expect the particles to undergo α collisions per unit time in average and all the correlations to be negligible. Therefore, depending on the initial distribution of positions and velocities in the 2dN -phase space, the 1-particle density f should satisfy a closed evolution equation where the inverse mean free path α measures the collision rate.

In the fast relaxation limit α → ∞, we then expect the system to relax towards local thermodynamic equilibrium, and the dynamics to be described by some macroscopic equations (depending on the observation time and length scales).

Figure 1. Fluid equations of hydrodynamics can be recovered directly from the microscopic system or in a two-step limit using Boltzmann's kinetic equation as a mesoscopic description. Note that these two procedures may lead to limiting equations with different transport coefficients since the kinetic equation describes only perfect gases (without excluded volume in the state relation).

One of the major difficulties to achieve this program using kinetic models as an intermediate description is to justify the low density limit α ≡ N ε d-1 on time intervals independent of α. Note that this step is also the most complicated one from the conceptual viewpoint as it should explain the appearance of irreversibility, and dissipation mechanisms.

The best result concerning the low density limit, which is due to Lanford in the case of hardspheres [START_REF] Lanford | Time evolution of large classical systems[END_REF] and King [START_REF] King | BBGKY hierarchy for positive potentials[END_REF] for more general potentials (see also [START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF][START_REF] Uchiyama | Derivation of the Boltzmann equation from particle dynamics[END_REF][START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] for a complete proof) is indeed valid only for short times, i.e. breaks down before any relaxation can be observed. The result may indeed be stated as follows [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] (see also [START_REF] Pulvirenti | On the validity of the Boltzmann equation for short range potentials[END_REF] for less restrictive assumptions on the potential Φ).

Theorem 1.1. Consider a system of N particles interacting

• either as hard-spheres of diameter ε • or as in (1.1) via a repulsive potential Φ, with compact support, radial and singular at 0, and such that the scattering of particles can be parametrized by their deflection angle.

Let f 0 : R 2d → R + be a continuous density of probability such that

f 0 exp( β 2 |v| 2 ) L ∞ (R d x ×R d v ) ≤ exp(-µ)
for some β > 0, µ ∈ R.

Assume that the N particles are initially distributed according to f 0 and "independent". Then, there exists some T * > 0 (depending only on β and µ) such that, in the Boltzmann-Grad limit N → ∞, ε → 0, N ε d-1 = α, the distribution function of the particles converges uniformly on [0, T * /α] × R 2d to the solution of the Boltzmann equation

(1.2) ∂ t f + v • ∇ x f = αQ(f, f ), Q(f, f )(v) := S d-1 ×R d [f (v * )f (v * 1 ) -f (v)f (v 1 )] b(v -v 1 , ν) dv 1 dν v * = v + ν • (v 1 -v) ν , v * 1 = v 1 -ν • (v 1 -v) ν
, with a locally bounded cross-section b depending on Φ implicitly, and with initial data f 0 . In the case of a hard-sphere interaction, the cross section is given by b

(v -v 1 , ν) = (v -v 1 ) • ν + .
Here, by "independent", we mean that the initial N -particle distribution satisfies a chaos property, namely that the correlations vanish asymptotically. Typically the distribution is obtained by factorization, and conditioning on energy surfaces (see [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] and references therein). In the case of hard-spheres for instance, one would have

f 0 N = Z -1 N f ⊗N 0 1 D N ε , with D N ε := (x 1 , v 1 , . . . , x N , v N ) ∈ T dN × R dN / ∀i = j , |x i -x j | > ε and f ⊗N 0 (x 1 , v 1 , . . . , x N , v N ) := N i=1 f 0 (x i , v i ) ,
while Z N normalizes the integral of f 0 N to 1.

The main difficulty to prove convergence for longer time intervals consists in ruling out the possibility of spatial concentrations of the density leading to some pathological collision process.

1.2. Linear regimes. In this paper, we overcome this difficulty by considering a good notion of fluctuation around global equilibrium for the system of interacting particles. In this way we get a complete derivation of the diffusion limit from the hard-sphere system in a linear regime. Of course, in this framework one cannot hope to retrieve a model for the full (nonlinear) gas dynamics, but -as far as we know -this is the very first result describing the Brownian motion as the limit of a deterministic classical system of interacting particles.

The main difficulty here is to justify the approximation by the linear Boltzmann equation (1.3)

∂ t ϕ α + v • ∇ x ϕ α = -α Lϕ α Lϕ α (v) := [ϕ α (v) -ϕ α (v )] M β (v 1 ) b(v -v 1 , ν) dv 1 dν M β (v) := β 2π d 2 exp - β 2 |v| 2 , β > 0 ,
for times diverging as α when α → ∞. Indeed, in the diffusive regime, the convergence of the Markov process associated to the linear Boltzmann operator L towards the Brownian motion is by now a classical result [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF].

Strategy and main results

A good notion of fluctuation is obtained by considering the motion of a tagged particle (or possibly a finite set of tagged particles) in a gas of N particles initially at equilibrium (or close to equilibrium), in the limit N → ∞.

2.1. The Lorentz gas. If the background particles are infinitely heavier than the tagged particle then the dynamics can be approximated by a Lorentz gas, i.e. by the motion of the tagged particle in a frozen background. The linear Boltzmann equation has been derived (globally in time) from the dynamics of a tagged particle in a low density Lorentz gas, meaning that • the obstacles are distributed randomly according to some Poisson distribution;

• the obstacles have no dynamics, in particular they do not feel the effect of collisions with the tagged particle. This problem, suggested by Lorentz [START_REF] Lorentz | Le mouvement des électrons dans les métaux[END_REF] at the beginning of the twentieth century to study the motion of electrons in metals, is the core of a number of works, and the corresponding literature includes a large variety of contributions. We do not intend to be exhaustive here and refer the reader to the book by Spohn [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF]Chapter 8] for a survey on this topic. We state one basic result due to Gallavotti [START_REF] Gallavotti | Statistical mechanics. A short treatise[END_REF] in the low density limit and then indicate some of the many important research directions.

Theorem 2.1. Consider randomly distributed scatterers with radius ε in R d according to a Poisson distribution of parameter αε 1-d . Let T t ε be the flow of a point particle reflected at the boundary of these scatterers. For a given continuous initial datum

f 0 ∈ L 1 ∩ L ∞ (R 2d ), we define f ε (t, x, v) := E[f 0 (T -t ε (x, v))] .
Then, for any time T > 0, f ε converges to the solution f of the linear Boltzmann equation (1.3), with hard-sphere cross-section, in

L ∞ ([0, T ], L 1 (R 2d )).
A refinement of this result can be found for instance in [START_REF] Spohn | The Lorentz process converges to a random flight process[END_REF] in terms of convergence of path measures (and not only of the mean density), as well as in [START_REF] Boldrighini | On the Boltzmann equation for the Lorentz gas[END_REF] where the convergence is proven for typical scatterer configurations (and not only in average).

These convergence statements lead naturally to various questions concerning

• the assumptions on the microscopic potential of interaction,

• the role of randomness for the distribution of scatterers,

• the long time behavior of the system, in particular the relaxation towards thermodynamic equilibrium and hydrodynamic limits.

The first point was addressed by Desvillettes, Pulvirenti and Ricci [START_REF] Desvillettes | The linear Boltzmann equation for long-range forces: a derivation from particles[END_REF][START_REF] Desvillettes | A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions[END_REF]. Their goal was to derive "singular " kinetic equations such as the linear Boltzmann equation without angular cut-off or the Fokker-Planck equation, from a system of particles with long-range interactions. They have obtained partial results in this direction, insofar as they can consider only asymptotically long-range interactions. Due to the fact that the range of the potential is infinite in the limit, the test particle interacts typically with infinitely many obstacles. Thus the set of bad configurations of the scatterers (such as the set of configurations yielding recollisions) preventing the Markov property of the limit must be estimated explicitly. Even though the long-range tails add a very small contribution to the total force for each typical scatterer distribution, the non grazing collisions generate an exponential instability making the two trajectories (with and without cut-off) very different. The complete derivation of the linear Boltzmann equation for long-range interactions is therefore still open.

It is often appropriate from a physical point of view to consider more general distributions of obstacles than the Poisson distribution. In particular, in the original problem of Lorentz, the atoms of metal are distributed on a periodic network. For the two-dimensional periodic Lorentz gas with fixed scatterer size, Bunimovich and Sinai [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] have shown the convergence, after a suitable time rescaling, of the tagged particle to a brownian motion. Their method relies on techniques from ergodic theory : it uses the fact that the mapping carrying a phase point on the boundary of a scatterer to the next phase point along its trajectory can be represented by a symbolic dynamics on a countable alphabet which is an ergodic Markov chain on a finite state space. Another important research direction, initiated by Golse, is to consider the periodic Lorentz gas in the Boltzmann-Grad limit. In this case there can be infinitely long free flight paths and the linear Boltzmann equation is no longer a valid limit [START_REF] Golse | On the periodic Lorentz gas and the Lorentz kinetic equation[END_REF][START_REF] Caglioti | On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas[END_REF][START_REF] Marklof | Kinetic transport in crystals[END_REF][START_REF] Marklof | The Boltzmann-Grad limit of the periodic Lorentz gas[END_REF], but the convergence toward a Brownian motion can be recovered after an appropriate superdiffusive rescaling [START_REF] Marklof | Superdiffusion in the periodic Lorentz gas[END_REF].

In [START_REF] Erdős | Quantum diffusion of the random Schrodinger evolution in the scaling limit[END_REF][START_REF] Erdős | Lecture notes on quantum Brownian motion[END_REF], Erdös, Salmhofer and Yau obtained the counterpart of the long time behavior for random quantum systems. Our approach is closer to their method than to the ones used for the periodic Lorentz gas (even though the setting of [START_REF] Erdős | Quantum diffusion of the random Schrodinger evolution in the scaling limit[END_REF] deals with a fixed random distribution of obstacles and a slightly different regime, known as weak coupling limit). Their proof is indeed based on a careful analysis of Duhamel's formula in combination with a renormalization of the propagator and stopping rules to control recollisions. We refer also to [START_REF] De Roeck | Diffusion of a massive quantum particle coupled to a quasi-free thermal medium[END_REF] for further developments on the quantum case.

2.2. Interacting gas of particles. We adopt here a different point of view, and consider a deterministic system of N hard-spheres, meaning that the tagged particle is identical to the particles of the background, interacting according to the same collision laws. In this paper, we will focus on the case d ≥ 2 (and refer to [START_REF] Szasz | Towards a unified dynamical theory of the Brownian particle in an ideal gas[END_REF] for results in the case d = 1). On the one hand, the problem seems more difficult than the Lorentz gas insofar as the background has its own dynamics, which is coupled with the tagged particle. But, on the other hand, pathological situations as described in [START_REF] Golse | On the periodic Lorentz gas and the Lorentz kinetic equation[END_REF][START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF][START_REF] Caglioti | On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas[END_REF] are not stable: because of the dynamics of the scatterers, we expect the situation to be better since some ergodicity could be retrieved from the additional degrees of freedom. In particular, there are invariant measures for the whole system, i.e. the system consisting in both the background and the tagged particle.

Here we shall take advantage of the latter property to establish global uniform a priori bounds for the distribution of particles, and more generally for all marginals of the N -particle distribution (see Proposition 4.1). This will be the key to control the collision process, and to prove (like in Kac's model [START_REF] Kac | Probability and related topics in physical sciences[END_REF] for instance) that dynamics for which a very large number of collisions occur over a short time interval, are of vanishing probability. Note that a similar strategy, based on the existence of the invariant measure, was already used by van Beijeren, Lanford, Lebowitz and Spohn [START_REF] Van Beijeren | Equilibrium Time Correlation Functions in the Low Density Limit[END_REF][START_REF] Lebowitz | Steady state self-diffusion at low density[END_REF] to derive the linear Boltzmann equation for long times.

Let us now give the precise framework of our study. As explained above, the idea is to improve Lanford's result by considering fluctuations around some global equilibrium. Locally the Nparticle distribution f N should therefore look like a conditioned tensorized Maxwellian.

In the sequel, we shall focus on the case of hard-sphere dynamics (with mass m = 1) to avoid technicalities due to artificial boundaries and cluster estimates. We shall further restrict our attention to the case when the domain is periodic

D = T d = [0, 1] d (d ≥ 2).
The microscopic model is therefore given by the following system of ODEs:

(2.1)

dx i dt = v i , dv i dt = 0 as long as |x i (t) -x j (t)| > ε for 1 ≤ i = j ≤ N ,
with specular reflection after a collision (2.2)

v i (t + ) = v i (t -) - 1 ε 2 (v i -v j ) • (x i -x j )(x i -x j )(t -) v j (t + ) = v j (t -) + 1 ε 2 (v i -v j ) • (x i -x j )(x i -x j )(t -)      if |x i (t) -x j (t)| = ε .
In the following we denote, for 1 ≤ i ≤ N , z i := (x i , v i ) and Z N := (z 1 , . . . , z N ). With a slight abuse we say that Z N belongs to T dN × R dN if X N := (x 1 , . . . , x N ) belongs to T dN and V N := (v 1 , . . . , v N ) to R dN . Recall that the phase space is denoted by

D N ε := Z N ∈ T dN × R dN / ∀i = j , |x i -x j | > ε .
We now distinguish pre-collisional configurations from post-collisional ones by defining for indexes 1

≤ i = j ≤ N ∂D N ± ε (i, j) := Z N ∈ T dN × R dN / |x i -x j | = ε , ±(v i -v j ) • (x i -x j ) > 0 and ∀(k, ) ∈ [1, N ] \ {i, j} 2 , |x k -x | > ε .
Given Z N on ∂D N + ε (i, j), we define Z * N ∈ ∂D N - ε (i, j) as the configuration having the same positions (x k ) 1≤k≤N , the same velocities (v k ) k =i,j for non interacting particles, and the following pre-collisional velocities for particles i and j

v * i := v i - 1 ε 2 (v i -v j ) • (x i -x j )(x i -x j ) v * j := v j + 1 ε 2 (v i -v j ) • (x i -x j )(x i -x j ) .
Defining the Hamiltonian

H N (V N ) := 1 2 N i=1 |v i | 2 ,
we consider the Liouville equation in the 2N d-dimensional phase space

D N ε (2.3) ∂ t f N + {H N , f N } = 0
with specular reflection on the boundary, meaning that if

Z N belongs to ∂D N + ε (i, j) then (2.4) f N (t, Z N ) = f N (t, Z * N )
. We recall, as shown in [START_REF] Alexander | The Infinite Hard Sphere System[END_REF] for instance, that the set of initial configurations leading to illdefined characteristics (due to clustering of collision times, or collisions involving more than two particles) is of measure zero in D N ε . Define the Maxwellian distribution by (2.5)

M ⊗s β (V s ) := s i=1 M β (v i ) and M β (v) := β 2π d 2 exp - β 2 |v| 2 .
An obvious remark is that M β is a stationary solution of (1.2), and any function of the energy f N ≡ F (H N ) is a stationary solution of the Liouville equation (2.3). In particular, for β > 0, the Gibbs measure with distribution in T dN × R dN defined by

(2.6) M N,β (Z N ) := 1 Z N β 2π dN 2 exp(-βH N (V N )) 1 D N ε (Z N ) = 1 Z N 1 D N ε (Z N )M ⊗N β (V N )
where the partition function Z N is the normalization factor (2.7)

Z N := T dN ×R dN 1 D N ε (Z N )M ⊗N β (V N ) dZ N = T dN 1≤i =j≤N 1 |x i -x j |>ε dX N ,
is an invariant measure for the gas dynamics.

In order to obtain the convergence for long times, a natural idea is to "weakly" perturb the equilibrium state M N,β , by modifying the distribution of one particle. In other words, we shall describe the dynamics of a tagged particle in a background initially at equilibrium.

Actually this is the reason for placing the study in a bounded domain, in order for M N,β to be integrable in the whole phase space. Moreover we have restricted our attention to the case of a torus in order to avoid pathologies related to boundary effects, and complicated free dynamics.

The strategy of perturbating M N,β is classical in probability theory; following this strategy • we lose asymptotically the nonlinear coupling: we thus expect to get a linear equation for the distribution of the tagged particle; • we also lose the feedback of the tagged particles on the background: since this background is constituted of N 1 indistinguishable particles, the momentum and energy exchange with the tagged particle has a very small effect on each one of these indistinguishable particles and thus does not modify on average the background distribution. As a consequence, the limiting equation for the distribution of the tagged particle should be non conservative. What we shall actually prove is that the limiting dynamics is governed by the linear Boltzmann equation (1.3) with hard-sphere cross-section.

Main results.

For the sake of simplicity, we consider only one tagged particle which will be labeled by 1 with coordinates z 1 = (x 1 , v 1 ). The initial data is a perturbation of the equilibrium density (2.6) only with respect to the position x 1 of the tagged particle. Consider ρ 0 a continuous density of probability on T d and define

(2.8) f 0 N (Z N ) := M N,β (Z N )ρ 0 (x 1 ) . Note that the distribution f 0 N is normalized by 1 in L 1 (T dN × R dN
) thanks to the translation invariance of T d and that

T d ρ 0 (x)dx = 1.
The main result of our study is the following statement.

Theorem 2.2. Consider the initial distribution f 0 N defined in (2.8). Then the distribution f

(1) N (t, x, v) of the tagged particle is close to M β (v)ϕ α (t, x, v), where ϕ α (t, x, v) is the solution of the linear Boltzmann equation (1.3) with initial data ρ 0 (x 1 ) and hard-sphere cross section. More precisely, for all t > 0 and all α > 1, in the limit N → ∞, N ε d-1 α -1 = 1, one has (2.9) f

(1)

N (t, x, v) -M β (v)ϕ α (t, x, v) L ∞ (T d ×R d ) ≤ C tα (log log N ) A-1 A A 2 A-1
, where A ≥ 2 can be taken arbitrarily large, and C depends on A, β, d and ρ 0 L ∞ . In [START_REF] Van Beijeren | Equilibrium Time Correlation Functions in the Low Density Limit[END_REF][START_REF] Lebowitz | Steady state self-diffusion at low density[END_REF], the linear Boltzmann equation was derived for any time t > 0 (independent of N ). In comparison, our approach leads to quantitative estimates on the convergence up to times diverging when N → ∞. As we shall see, this is the key to derive the diffusive limit in Theorem 2.3. Theorem 2.2 proves that the linear Boltzmann equation is a good asymptotics of the hard-sphere dynamics, even for large concentrations α and long times t. It further provides a rather good estimate on the approximation error. Up to a suitable rescaling of time, we can therefore obtain diffusive limits.

In the macroscopic limit, the trajectory of the tagged particle is defined by

(2.10) Ξ(τ ) := x 1 ατ ∈ T d .
The distribution of Ξ(τ ) is given by f

N (ατ, x, v). In the following, τ represents the macroscopic time scale. Theorem 2.3. Consider N hard spheres on the space T d ×R d , initially distributed according to f 0 N defined in (2.8). Assume that ρ 0 belongs to C 0 (T d ). Then the distribution f

N (ατ, x, v) remains close for the L ∞ -norm to ρ(τ, x)M β (v) where ρ(τ, x) is the solution of the linear heat equation

(2.11) ∂ τ ρ -κ β ∆ x ρ = 0 in T d , ρ |τ =0 = ρ 0 ,
and the diffusion coefficient κ β is given by

κ β := 1 d R d vL -1 v M β (v)dv,
where L is the linear Boltzmann operator (1.3) and L -1 is its pseudo-inverse defined on (Ker L) ⊥ (see also (6.8)). More precisely,

(2.12) f

(1)

N (ατ, x, v) -ρ(τ, x)M β (v) L ∞ ([0,T ]×T d ×R d ) → 0 in the limit N → ∞, with α = N ε d-1
going to infinity much slower than √ log log N . In the same asymptotic regime, the process Ξ(τ ) = x 1 (ατ ) associated with the tagged particle converges in law towards a Brownian motion of variance κ β , initially distributed under the measure ρ 0 .

The Boltzmann-Grad scaling α = N ε d-1 is chosen such that the mean free path is of order 1/α, i.e. that a particle has on average α collisions per unit time. This explains why in (2.10), the position of the particle is not rescaled. Indeed over a time scale ατ a particle will encounter α 2 τ collisions which is the correct balance for a diffusive limit. In other words, one can think of α as a parameter tuning the density of the background particles. The positions and velocities are not rescaled with α and are always at the macroscopic scale.

2.4. Generalizations. For the sake of clarity, Theorem 2.3 has been stated in the simplest framework. We mention below several extensions which can be deduced in a straightforward way from the proof of Theorem 2.3.

Several tagged particles :

The dynamics of a finite number of tagged particles can be followed and one can show that asymptotically, they converge to independent Brownian motions. This gives an answer to a conjecture raised by Lebowitz and Spohn [START_REF] Lebowitz | Microscopic Basis for Fick's Law for Self-Diffusion[END_REF] on the diffusion of colored particles in a fluid.

Interaction potential :

Following the arguments in [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF][START_REF] Pulvirenti | On the validity of the Boltzmann equation for short range potentials[END_REF], the behavior of a tagged particle in a gas with an interaction potential can also be treated.

Initial data :

The perturbation on the initial particle could depend on z 1 = (x 1 , v 1 ) instead of depending only on the position x 1 . The comparison argument to the linear Boltzmann equation is identical, but the derivation of the diffusive behavior in Section 6.1 should be modified to show the relaxation of the velocity to a Maxwellian at the initial stage (see Remark 6.2). By considering an initial data of the form (2.13)

ρ 0 α (x 1 ) = α dζ ρ 0 α ζ x 1 with ζ 1
the tagged particle localizes when α goes to infinity. The analysis can be extended to this class of initial data and leads, in the macroscopic limit, to a Brownian motion starting initially from a Dirac mass.

Scalings :

We have chosen here to work with macroscopic variables (x, v), i.e. to rescale the particle concentration of the background and to dilate the time with a factor α. However, the diffusive limit can be obtained by many other equivalent scalings involving the space variable. In particular, one could have considered a domain [0, λ] d with a size λ growing and a Boltzmann-Grad scaling (N/λ d )ε d-1 = 1. Rescaling space by a factor λ and time by λ 2 log log N would have led to the same diffusive limit. In fact, one only needs the Knudsen number to be small and of the same order as the Strouhal number [START_REF] Bardos | Fluid Dynamic Limits of the Boltzmann Equation I[END_REF][START_REF] Saint-Raymond | Hydrodynamic limits of the Boltzmann equation[END_REF]. 2.5. Structure of the paper. Theorem 2.3 is a consequence of Theorem 2.2, as explained in Section 6. The core of our study is therefore the proof of Theorem 2.2, which relies on a comparison of the particle system to a limit system known as Boltzmann hierarchy. This hierarchy is obtained formally in Section 3 from the hierarchy of equations satisfied by the marginals of f N , known as the BBGKY hierarchy (which is introduced in Section 3). Section 4 is devoted to the control of the branching process that can be associated with the hierarchies, and in particular with the elimination of super-exponential trees; the specificity of the linear framework is crucial in this step, as it makes it possible to compare the solution with the invariant measure globally in time. The actual proof of the convergence of the BBGKY hierarchy towards the Boltzmann hierarchy, on times diverging with N , can be found in Section 5. Some more technical estimates are postponed to Appendix A and B.

Formal derivation of the low density limit

Our starting point to study the low density limit is the Liouville equation (2.3) and its projection on the first marginal

f (1) N (t, z 1 ) := f N (t, Z N )dz 2 . . . dz N .
Since it does not satisfy a closed equation, we have to consider the whole BBGKY hierarchy (see Paragraph 3.1). The main difference with the usual strategy to prove convergence is that the symmetry is partially broken due to the fact that one particle is distinguished from the others. In other words f N |t=0 is symmetric with respect to z 2 , . . . z N but not to z 1 , and this property is preserved by the dynamics. More precisely we shall see that the specific form of the initial data (see Paragraph 3.2) implies that asymptotically we have the following closure

f (2) N (t, z 1 , z 2 ) := f N (t, Z N )dz 3 . . . dz N ∼ f (1) N (t, z 1 )M β (v 2 ) ∼ ϕ α (t, z 1 )M β (v 1 )M β (v 2 )
where ϕ α satisfies the linear Boltzmann equation (1.3) with initial data ρ 0 . Thus the limiting hierarchy reduces to the linear Boltzmann equation (see Paragraph 3.3).

3.1.

The series expansion. The quantities we shall consider are the marginals

f (s) N (t, Z s ) := f N (t, Z N )dz s+1 . . . dz N so f (1)
N is exactly the distribution of the tagged particle, and f (s) N is the correlation between this tagged particle and (s -1) particles of the background. A formal computation based on Green's formula leads to the following BBGKY hierarchy for s < N

(3.1) (∂ t + s i=1 v i • ∇ x i )f (s) N (t, Z s ) = α C s,s+1 f (s+1) N (t, Z s )
on D s ε , with the boundary condition as in (2.4)

f (s) N (t, Z s ) = f (s) N (t, Z * s ) on ∂D s+ ε (i, j) .
The collision term is defined by

(3.2) C s,s+1 f (s+1) N (Z s ) := (N -s)ε d-1 α -1 × s i=1 S d-1 ×R d f (s+1) N (. . . , x i , v * i , . . . , x i + εν, v * s+1 ) (v s+1 -v i ) • ν + dνdv s+1 - s i=1 S d-1 ×R d f (s+1) N (. . . , x i , v i , . . . , x i + εν, v s+1 ) (v s+1 -v i ) • ν - dνdv s+1
where S d-1 denotes the unit sphere in R d . Note that the collision integral is split into two terms according to the sign of (v i -v s+1 ) • ν and we used the trace condition on ∂D N ε to express all quantities in terms of pre-collisional configurations.

The closure for s = N is given by the Liouville equation (2.3). Note that the classical symmetry arguments used to establish the BBGKY hierarchy, i.e. the evolution equations for the marginals f (s) N (t, Z s ), only involve the particles we add by collisions to the sub-system Z s under consideration. In particular, the equation in the BBGKY hierarchy will not be modified at all since -by convention -the tagged particle is labeled by 1 and always belongs to the sub-system under consideration.

Given the special role played by the initial data (which is the reference to determine the notion of pre-collisional and post-collisional configurations), it is then natural to express solutions of the BBGKY hierarchy in terms of a series of operators applied to the initial marginals. The starting point in Lanford's proof is therefore the iterated Duhamel formula

(3.3) f (s) N (t) = N -s n=0 α n t 0 t 1 0 . . . t n-1 0 S s (t -t 1 )C s,s+1 S s+1 (t 1 -t 2 )C s+1,s+2 . . . S s+n (t n )f (s+n) N (0) dt n . . . dt 1 ,
where S s denotes the group associated to free transport in D s ε with specular reflection on the boundary.

To simplify notations, we define the operators Q s,s (t) = S s (t) and for n ≥ 1 (3.4)

Q s,s+n (t) := t 0 t 1 0 . . . t n-1 0 S s (t -t 1 )C s,s+1 S s+1 (t 1 -t 2 )C s+1,s+2 . . . S s+n (t n ) dt n . . . dt 1 so that (3.5) f (s) N (t) = N -s n=0 α n Q s,s+n (t)f (s+n) N (0) . Remark 3.1.
It is not obvious that formula (3.5) makes sense since the transport operator S s+1 is defined only for almost all initial configurations, and the collision operator C s,s+1 is defined by some integrals on manifolds of codimension 1. This fact is analyzed in [START_REF] Simonella | Evolution of correlation functions in the hard sphere dynamics[END_REF] and in the erratum of [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF]. In the following, we will rely on the estimates on the collision operator derived in [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF].

3.2.

Asymptotic factorization of the initial data. The effect of the exclusion in the equilibrium measure vanishes when ε goes to 0 and the particles become asymptotically independent in the following sense.

Proposition 3.2. Given β > 0, there is a constant C > 0 such that for any fixed s ≥ 1, the marginal of order s

(3.6) M (s) N,β (Z s ) := M N,β (Z N ) dz s+1 . . . dz N satisfies, as N → ∞ in the scaling N ε d-1 ≡ α 1/ε, (3.7) M (s) N,β -M ⊗s β 1 D s ε ≤ C s εα M ⊗s β
where the Maxwellian distribution M ⊗s β was introduced in (2.5). The proof of Proposition 3.2, by now classical, is recalled in Appendix A for the sake of completeness. As a consequence of Proposition 3.2, the initial data is asymptotically close to a product measure: the following result is a direct corollary of Proposition 3.2. Proposition 3.3. For the initial data f 0 N given in (2.8), define the marginal of order s f 0(s)

N (Z s ) := f 0 N (Z N ) dz s+1 . . . dz N = ρ 0 (x 1 )M (s) N,β (Z s ) .
There is a constant

C > 0 such that as N → ∞ in the scaling N ε d-1 = α 1/ε f 0(s) N -g 0(s) 1 D s ε ≤ C s εαM ⊗s β ρ 0 L ∞ ,
where g 0(s) is defined by

(3.8) g 0(s) (Z s ) := ρ 0 (x 1 )M ⊗s β (V s ) . 3.3.
The limiting hierarchy and the linear Boltzmann equation. To obtain the Boltzmann hierarchy we start with the expansion (3.5) and compute the formal limit of the collision operator Q s,s+n when ε goes to 0. Recalling that (

N -s)ε d-1 α -1 ∼ 1, it is given by Q 0 s,s+n (t) := t 0 t 1 0 . . . t n-1 0 S 0 s (t -t 1 )C 0 s,s+1 S 0 s+1 (t 1 -t 2 )C 0 s+1,s+2 . . . S 0 s+n (t n ) dt n . . . dt 1
where S 0 s denotes the free flow of s particles on T ds × R ds , and C 0 s,s+1 are the limit collision operators defined by (3.9)

C 0 s,s+1 g (s+1) (Z s ) := s i=1 g (s+1) (. . . , x i , v * i , . . . , x i , v * s+1 ) (v s+1 -v i ) • ν + dνdv s+1 - s i=1 g (s+1) (. . . , x i , v i , . . . , x i , v s+1 ) (v s+1 -v i ) • ν - dνdv s+1 .
Then the iterated Duhamel formula for the Boltzmann hierarchy takes the form

(3.10) ∀s ≥ 1 , g (s) α (t) = n≥0 α n Q 0 s,s+n (t)g 0(s+n) .
Remark 3.4. In the Boltzmann hierarchy, the collision operators are defined by integrals on manifolds of codimension d, so we shall require that the functions g (s)

α s≥1 are continuous, which is possible since free transport preserves continuity on

T d × R d .
Consider the initial data (3.8). Then the family (g

(s) α ) s≥1 defined by (3.11) g (s) α (t, Z s ) := ϕ α (t, z 1 )M ⊗s β (V s )
is a solution to the Boltzmann hierarchy with initial data g 0(s) since ϕ α satisfies the linear Boltzmann equation (1.3) with initial data ρ 0 . We insist that the g (s)

α are not defined as the marginals of some N -particle density.

Remark 3.5. Note that the estimates established in the next section imply actually that (g

(s)
α ) s≥1 is the unique solution to the Boltzmann hierarchy (see [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF]). Furthermore the maximum principle for the linear Boltzmann equation leads to the following estimate sup

t≥0 ϕ α (t, z 1 ) ≤ ρ 0 L ∞ .
In the following for the sake of simplicity we write g α := g

α .

Control of the branching process

The restriction on the time of validity T * /α of Lanford's convergence proof (determined by a weighted norm of the initial data) is based on the elimination of "pathological" collision trees, defined by a too large number of branches created in the time interval [0, T * /α] (typically greater than n ε = O(| log ε|), see [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] for a quantitative estimate of the truncation parameter). Here the global bound coming from the maximum principle will enable us to iterate this truncation process on any time interval.

4.1.

A priori estimates coming from the maximum principle. For initial data as (2.8), uniform a priori bounds can be obtained using only the maximum principle for the Liouville equation (2.3).

f (s) N (t, Z s ) := f N (t, Z N ) dz s+1 . . . dz N .
Then, for any s ≥ 1, the following bounds hold uniformly with respect to time

(4.2) sup t f (s) N (t, Z s ) ≤ M (s) N,β (Z s ) ρ 0 L ∞ ≤ C s M ⊗s β (V s ) ρ 0 L ∞ ,
for some C > 0, provided that αε 1.

Note here that although the variable z 1 does not play at all a symmetric role with respect to z 2 , . . . z N , the upper bound (4.2) does not see this asymmetry.

Proof. One has immediately from (2.8) that

f 0 N (Z N ) = M N,β (Z N )ρ 0 (x 1 ) ≤ M N,β (Z N ) ρ 0 L ∞ .
Since the maximum principle holds for the Liouville equation (2.3), and as the Gibbs measure M N,β is a stationary solution, we get for all t ≥ 0

f N (t, Z N ) ≤ M N,β (Z N ) ρ 0 L ∞ .
The inequalities for the marginals follow by integration and Proposition 3.2.

Continuity estimates for the collision operators.

To get uniform estimates with respect to N , the usual strategy is to use some Cauchy-Kowalewski argument. In the following we shall denote by |Q| s,s+n the operator obtained by summing the absolute values of all elementary contributions

|Q| s,s+n (t) := t 0 t 1 0 . . . t n-1 0 S s (t -t 1 ) |C s,s+1 | S s+1 (t 1 -t 2 ) |C s+1,s+2 | . . . S s+n (t n ) dt n . . . dt 1 and similarly for |Q 0 | s,s+n |Q 0 | s,s+n (t) := t 0 t 1 0 . . . t n-1 0 S 0 s (t -t 1 ) |C 0 s,s+1 | S 0 s+1 (t 1 -t 2 ) |C 0 s+1,s+2 | . . . S 0 s+n (t n ) dt n . . . dt 1 where |C s,s+1 |f (s+1) N (Z s ) := (N -s)ε d-1 α -1 s i=1 S d-1 ×R d f (s+1) N (. . . , x i , v * i , . . . , x i + εν, v * s+1 ) (v s+1 -v i ) • ν + dνdv s+1 + (N -s)ε d-1 α -1 s i=1 S d-1 ×R d f (s+1) N (. . . , x i , v i , . . . , x i + εν, v s+1 ) (v s+1 -v i ) • ν - dνdv s+1 and |C 0 s,s+1 |g (s+1) (Z s ) := s i=1 g (s+1) (. . . , x i , v * i , . . . , x i , v * s+1 ) (v s+1 -v i ) • ν + dνdv s+1 + s i=1 g (s+1) (. . . , x i , v i , . . . , x i , v s+1 ) (v s+1 -v i ) • ν - dνdv s+1 .
For λ > 0 and k ∈ N * , we define X ε,k,λ the space of measurable functions f k defined almost everywhere on

D k ε such that f k ε,k,λ := supess Z k ∈D k ε f k (Z k ) exp λH k (Z k ) < ∞
, and similarly X 0,k,λ is the space of continuous functions g k defined on T dk × R dk such that Similarly for all g s+n in X 0,s+n,λ , |Q 0 | s,s+n (t)g s+n belongs to X 0,s, λ 2 and

g k 0,k,λ := sup Z k ∈T dk ×R dk g k (Z k ) exp λH k (Z k ) < ∞ . Lemma 4.2.
(4.4) |Q 0 | s,s+n (t)g s+n 0,s, λ 2 ≤ e s-1 C d t λ d+1 2 n g s+n 0,s+n,λ .
Proof. Estimate (4.3) is simply obtained from the fact that the transport operators preserve the weighted norms, along with the continuity of the elementary collision operators. From the erratum of [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF], we get the following statements • the transport operators satisfy the identities

S k (t)f k ε,k,λ = f k ε,k,λ S 0 k (t)g k 0,k,λ = g k 0,k,λ .
• the collision operators satisfy the following bounds in the Boltzmann-Grad scaling

N ε d-1 ≡ α S k (-t) |C k,k+1 | S k+1 (t)f k+1 (Z k ) ≤ C d λ -d 2 kλ -1 2 + 1≤i≤k |v i | exp (-λH k (Z k )) f k+1 ε,k+1,λ
almost everywhere on R t × D k ε , for some C d > 0 depending only on d, and (4.5)

|C 0 k,k+1 | g k+1 (Z k ) ≤ C d λ -d 2 kλ -1 2 + 1≤i≤k |v i | exp (-λH k (Z k )) g k+1 0,k+1,λ , on T dk × R dk .
The result then follows from piling together those inequalities (distributing the exponential weight evenly on each occurence of a collision term). We notice that by the Cauchy-Schwarz inequality,

1≤i≤k |v i | exp - λ 4n 1≤j≤k |v j | 2 ≤ k 2n λ 1 2   1≤i≤k λ 2n |v i | 2 exp - λ 2n 1≤j≤k |v j | 2   1/2 ≤ 2nk eλ 1/2 ≤ 2 eλ (s + n) ,
with k ≤ s + n in the last inequality. Each collision operator gives therefore a loss of Cλ -(d+1)/2 (s + n) together with a loss on the exponential weight, while the integration with respect to time provides a factor t n /n!. By Stirling's formula, we have

(s + n) n n! ≤ exp n log n + s n + n ≤ exp(s + n) .
That proves the first statement in the lemma. The same arguments give the counterpart for the Boltzmann collision operator.

4.3.

Collision trees of controlled size. For general initial data (in particular, for chaotic initial data), the proof of Lanford's convergence result then relies on two steps: (i) a short time bound for the series expansion (3.5) expressing the correlations of the system of N particles and a similar bound for the corresponding quantities associated with the Boltzmann hierarchy; (ii) the termwise convergence of each term of the series. However after a short time (depending on the initial data), the question of the convergence of the series (3. 5) is still open. One of the difficulties to prove this convergence is to take into account the cancellations between the gain and loss terms of the collision operators. These cancellations are neglected in Lanford's strategy. Here we assume that the BBGKY initial data takes the form (2.8) and the Boltzmann initial data takes the form (3.8), and we shall take advantage of the control by stationary solutions (the existence of which is obviously related to these cancellations) given by Proposition 4.1 to obtain a lifespan which does not depend on the initial data. Indeed, we have thanks to Propositions 3.2 and 4.1 provided that αε 1

f (k) N (t) ε,k,β = supess Z k ∈D k ε f (k) N (t, Z k ) exp βH k (Z k ) ≤ sup Z k ∈D k ε M (k) N,β (Z k ) exp βH k (Z k ) ρ 0 L ∞ ≤ C k sup Z k ∈D k ε M ⊗k β (V k ) exp βH k (Z k ) ρ 0 L ∞ .
Thus for all t ∈ R,

(4.6) f (k) N (t) ε,k,β ≤ C k β 2π kd/2 ρ 0 L ∞ .
Similarly for the initial data for the Boltzmann hierarchy defined in (3.8), by Remark 3.5 the solution (3.11) of the evolution is bounded by

(4.7) g (k) α (t) 0,k,β ≤ β 2π kd/2 ρ 0 L ∞ .
Moreover we shall use a truncated series expansion instead of (3.5) and (3.10). Let us fix a (small) parameter h > 0 and a sequence {n k } k≥1 of integers to be tuned later. We shall study the dynamics up to time t := Kh for some large integer K, by splitting the time interval [0, t] into K intervals, and controlling the number of collisions on each interval. In order to discard trajectories with a large number of collisions in the iterated Duhamel formula (3.5), we define collision trees "of controled size" by the condition that they have strictly less than n k branch points on the interval [t -kh, t -(k -1)h]. Note that by construction, the trees are actually followed "backwards", from time t (large) to time 0.

t h Figure 2. Suppose n k = A k with A = 2.
Each collision is represented by a circle from which 2 trajectories emerge. The tree including the three extra collisions in dotted lines occurring during [t -2h, t -h] is not a good collision tree and in our procedure, it would be truncated at time t -2h. The tree without the dotted lines is a good collision tree with t = 4h : the number of collisions during the k th -time interval is less than

n k -1 = A k -1.
As we are interested only in the asymptotic behaviour of the first marginal, we start by using (3.3) with s = 1, during the time interval [t -h, t]: iterating Duhamel's formula up to time t -h instead of time 0, we have

(4.8) f (1)
N (t) = n 1 -1 j 1 =0 α j 1 -1 Q 1,1+j 1 (h)f (j 1 ) N (t -h) + R 1,n 1 (t -h, t) ,
where R 1,n 1 accounts for at least n 1 collisions

R 1,n 1 (t , t) := N -1 p=n 1 α p Q 1,p+1 (t -t )f (p+1) N (t ) .
More generally we define R k,n as follows

R k,n (t , t) := N -k p=n α p Q k,k+p (t -t )f (k+p) N (t ) .
The term R k,n (t , t) accounts for trajectories originating at k points at time t, and involving at least n collisions during the time-span t -t . The idea is that if n is large then such a behaviour should be atypical and R k,n (t , t) should be negligible.

The first term on the right-hand side of (4.8) can be broken up again by iterating the Duhamel formula on the time interval [t-2h, t-h] and truncating the contributions with more than n 2 collisions: this gives f

(1)

N (t) = n 1 -1 j 1 =0 n 2 -1 j 2 =0 α j 1 +j 2 Q 1,1+j 1 (h)Q 1+j 1 ,1+j 1 +j 2 (h) f (1+j 1 +j 2 ) N (t -2h) + R 1,n 1 (t -h, t) + n 1 -1 j 1 =0 α j 1 Q 1,j 1 +1 (h)R j 1 +1,n 2 (t -2h, t -h) .
Iterating this procedure K times and truncating the trajectories with at least n k collisions during the time interval [t -kh, t -(k -1)h], leads to the following expansion (4.9) f

(1)

N (t) = f (1,K) N (t) + R K N (t)
, where denoting J 0 := 1 and

J k := 1 + j 1 + • • • + j k , (4.10) f (1,K) N (t) := n 1 -1 j 1 =0
. . .

n K -1 j K =0 α J K -1 Q 1,J 1 (h)Q J 1 ,J 2 (h) . . . Q J K-1 ,J K (h) f 0(J K ) N and R K N (t) := K k=1 n 1 -1 j 1 =0
. . .

n k-1 -1 j k-1 =0 α J k-1 -1 Q 1,J 1 (h) . . . Q J k-2 ,J k-1 (h) R J k-1 ,n k (t -kh, t -(k -1)h) .
By an appropriate choice of the sequence {n k }, we are going to show that the main contribution to the density f

N (t) is given by f

(1,K) N

(t) and that R K N (t) vanishes asymptotically.

Next as in (4.10) we can write a truncated expansion for g α (see (3.11)) as follows:

(4.11)

g α (t) = g (1,K) α (t) + R 0,K α (t)
, where with notation (3.8) and (3.11), (4.12)

g (1,K) α (t) := n 1 -1 j 1 =0 . . . n K -1 j K =0 α J K -1 Q 0 1,J 1 (h)Q 0 J 1 ,J 2 (h) . . . Q 0 J K-1 ,J K (h) g 0(J K ) α and R 0,K α (t) := K k=1 n 1 -1 j 1 =0
. . . We shall choose the latter option. Proposition 4.3. Under the assumptions of Theorem 2.2, the following holds. Let A ≥ 2 be given and define n k := A k , for k ≥ 1. Then there exist c, C, γ 0 > 0 depending on d, A and β such that for any t > 1 and any γ ≤ γ 0 , choosing (4. [START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF] h ≤ cγ α A/(A-1) t 1/(A-1) and K = t/h integer we get

n k-1 -1 j k-1 =0 α J k-1 -1 Q 0 1,J 1 (h) . . . Q 0 J k-2 ,J k-1 (h) R 0 J k-1 ,n k (t -kh, t -(k -1)h) with R 0 k,n (t , t) := p≥n α p Q 0 k,k+p (t -t )g (k+p) α (t ) .
(4.14) R K N (t) L ∞ (T d ×R d ) + R 0,K α (t) L ∞ (T d ×R d ) ≤ Cγ A ρ 0 L ∞ .
Proof. We are going to bound

Q 1,J 1 (h) . . . Q J k-2 ,J k-1 (h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d )
for each term in the remainder R K N . The exact distribution of collisions in the last k -1 intervals is not needed and it is enough to estimate directly

|Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) .
Applying Lemma 4.2, one has (denoting generically by C d any constant depending only on d)

|Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ C d (k -1)h β (d+1)/2 J k-1 -1 R J k-1 ,n k (t -kh, t -(k -1)h) ε,J k-1 ,β/2 .
Then arguing as in the proof of Lemma 4.2, one can write

α J k-1 -1 |Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ N -J k-1 p=n k C d α(k -1)h β (d+1)/2 J k-1 -1 C d αh β (d+1)/2 p sup t≥0 f (J k-1 +p) N (t) ε,J k-1 +p,β ≤ ρ 0 L ∞ β d 2 (αt) J k-1 -1 N -J k-1 p=n k C d √ β J k-1 +p-1 (αh) p ,
thanks to (4.6) and recalling that (k -1)h ≤ t. Assuming from now on that (4.15)

C d αh √ β < 1 2
we find (4.16)

α J k-1 -1 |Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ ρ 0 L ∞ β d 2 (αt) J k-1 -1 C d √ β J k-1 +n k -1 (αh) n k . Note that N j := 1 + n 1 + • • • + n j = A j+1 -1 A-1 ≤ 1 A-1 n j+1 . Then, since J k-1 ≤ N k-1 , one has, for some appropriate constant C(d, β), α J k-1 -1 |Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ β d/2 exp A k log C(d, β) + 1 A -1 log(αt) + log(αh) ρ 0 L ∞ .
Therefore, choosing 1) , which is compatible with (4.15) as soon as γ is small enough one has (4.17)

h ≤ γ C(d, β) α A/(A-1) t 1/(A-
α J k-1 -1 |Q| 1,J k-1 ((k -1)h) R J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ β d/2 exp A k log γ ρ 0 L ∞ .
This implies

R K N L ∞ (T d ×R d ) ≤ β d/2 K k=1 k i=1 n i exp A k log γ ρ 0 L ∞ ≤ β d/2 K k=1 exp k(k + 1) log(A) + A k log γ ρ 0 L ∞ ≤ C A β d/2 K k=1 exp Ak log γ ρ 0 L ∞ ≤ C A β d/2 γ A ρ 0 L ∞
for γ sufficiently small, where C A is a constant depending on A. Thus, we get the first part of (4.14)

R K N L ∞ (T d ×R d ) ≤ Cγ A ρ 0 L ∞ .
The argument is identical in the case of the Boltzmann hierarchy:

α J k-1 -1 |Q| 0 1,J k-1 ((k -1)h) R 0 J k-1 ,n k (t -kh, t -(k -1)h) L ∞ (T d ×R d ) ≤ C d (k -1)αh β (d+1)/2 J k-1 -1 C d αh β (d+1)/2 n k sup t≥0 g (J k-1 +n k ) α (t) 0,J k-1 +n k ,β ≤ β d 2 C d √ β J k-1 +n k -1 (αt) J k-1 -1 (αh) n k ρ 0 L ∞ , hence finally R 0,K α L ∞ (T d ×R d ) ≤ C A β d/2 γ A ρ 0 L ∞
, and the proposition is proved.

Proof of the convergence

In this section, we conclude the proof of Theorem 2.2. Thanks to Proposition 4.3, we are reduced to studying f

(1,K) N -g (1,K) α
(introduced in (4.9), (4.11)) and to proving that the matching terms in the series f Throughout this section, the parameters are chosen such that (with the notation of Proposition 4.3)

(5.1) N ε d-1 = α 1 ε , A ≥ 2 , t > 1 , K = t h •
Each elementary term in the series f

(1,K) N and g

(1,K) α has a geometric interpretation as an integral over some pseudo-trajectories. As explained in [START_REF] Lanford | Time evolution of large classical systems[END_REF][START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF][START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF], in this formulation the characteristics associated with the operators S i (t i-1 -t i ) and S 0 i (t i-1 -t i ) are followed backwards in time between two consecutive times t i and t i-1 , and the collision terms (associated with C i,i+1 and C 0 i,i+1 ) are seen as source terms in which "additional particles" are "adjoined" to the system. The main heuristic idea is that the pseudo-trajectories associated to both hierarchies can be coupled precisely if no recollisions occur in the BBGKY hierarchy. The core of the proof will be to obtain an upper bound on the occurrence of recollisions and to show that their contribution is negligible. In order to prevent recollisions in the time interval [t i+1 , t i ], some bad sets in phase space must be removed. Following the approach developed in [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF], a geometrical control of the trajectories in the torus (stated in Lemma 5.2) enables us to define bad sets, outside of which the flow S between two collision times is the free flow S 0 (see Proposition 5.1). Finally, the geometric controls are used in Section 5.3 to obtain quantitative estimates on the collision integrals where those bad sets have been removed. 5.1. Reformulation in terms of pseudo-trajectories. We consider one term of the sum f

(1,K) N (t) in (4.10) and show how it can be interpreted in terms of pseudo-trajectories. Given the indices J = (j 1 , . . . , j K ), we set

F (1,K) N (J) (t, z 1 ) := Q 1,J 1 (h)Q J 1 ,J 2 (h) . . . Q J K-1 ,J K (h) f 0(J K ) N (5.2) = T J (h) dT S 1 (t -t 1 )C 1,2 S 2 (t 1 -t 2 )C 2,3 . . . S J K (t J K -1 )f 0(J K ) N
where the time integral is over the collision times T = (t 1 , . . . , t J K-1 ) taking values in

(5.3) T J (h) := T = (t 1 , . . . , t J K-1 ) t i < t i-1 and (t J k , . . . , t J k-1 +1 ) ∈ [t-kh, t-(k -1)h] .
In the following we denote by Ψ s the s-particle flow. Given z 1 = (x 1 , v 1 ) ∈ T d × R d and a time u ∈ [t 1 , t], we call z 1 (u) = Ψ 1 (u)z 1 the coordinates following the backward flow Ψ 1 of one particle. The first collision operator C 1,2 is interpreted as the adjunction at time t 1 of a new particle at x 1 (t 1 ) + εν 2 for a deflection angle ν 2 ∈ S d-1 and with a velocity v 2 ∈ R d . The new pair of particles Z 2 will be evolving according to the backward 2-particle flow Ψ 2 during the time interval [t 2 , t 1 ] starting at t 1 from

Z 2 (t 1 ) = (x 1 (t 1 ), v 1 ), (x 1 (t 1 ) + εν 2 , v 2 ) in the pre-collisional case (v 2 -v 1 ) • ν 2 < 0 Z 2 (t 1 ) = (x 1 (t 1 ), v * 1 ), (x 1 (t 1 ) + εν 2 , v * 2 ) in the post-collisional case (v 2 -v 1 ) • ν 2 > 0 , (5.4)
the latter case corresponding to the scattering.

Iterating this procedure, a branching process is built inductively by adding a particle labelled i + 1 at time t i to the particle z m i (t i ) where m i ≤ i is chosen randomly among the first i particles. Given a deflection angle ν i+1 and a velocity v i+1 , the velocity of the particles z m i and z i+1 at time t i are updated according to the pre-collisional or post-collisional rule as in (5.4)

         Z i+1 (t i ) = {z j (t i )} j =m i , (x m i (t i ), v m i (t i )), (x m i (t i ) + εν i+1 , v i+1 ) in the pre-collisional case (v i+1 (t i ) -v m i ) • ν i+1 < 0 Z i+1 (t i ) = {z j (t i )} j =m i , (x m i (t i ), v * m i (t i )), (x m i (t i ) + εν i+1 , v * i+1 ) in the post-collisional case (v i+1 (t i ) -v m i ) • ν i+1 > 0 .
Let Z i+1 denote the i + 1 components after the i th -collision. The evolution of Z i+1 follows the flow of the backward transport Ψ i+1 during the time interval [t i+1 , t i ]. From [START_REF] Simonella | Evolution of correlation functions in the hard sphere dynamics[END_REF] (see also Remark 3.1), one can check that Ψ i+1 is well defined up to a set of measure 0. In the following, we shall use the name collision to describe the creation of a particle and recollision if two particles collide in the flow Ψ i+1 .

To summarize, pseudo-trajectories do not involve physical particles. They are a geometric interpretation of the iterated Duhamel formula in terms of a branching process flowing backward in time and determined by

• the collision times T = (t 1 , . . . , t J K -1 ) which are interpreted as branching times

t t 1 t 2 t 3 t 4 = 0 z 1 x 1 (t 1 )
x 1 (0)

x 3 (0) x 2 (0) x 2 (t 2 )
x 4 (0) Figure 3. A collision tree is represented with 3 collisions. The velocities (v 1 , v 2 ) at time t 1 are pre-collisional and the first particle keeps its velocity v 1 after the collision. At time t 3 , the first particle is selected m 3 = 1 and the velocity v 1 is modified into v * 1 according to the post-collisional rule.

• the labels of the collision particles m = (m 1 , . . . , m J K -1 ) from which branching occurs and which take values in the set

M J := m = (m 1 , . . . , m J K -1 ) , 1 ≤ m i ≤ i
• the coordinates of the initial particle z 1 at time t • the velocities v 2 , . . . , v J K in R d and deflection angles ν 2 , . . . , ν J K in S d-1

1
for each additional particle.

The integral (5.2) can be evaluated by integrating f 0(J K ) N on the value of the pseudo-trajectories Z J K (0) at time 0

F (1,K) N (J) = m∈M J ε d-1 α J K -1 (N -1)! (N -J K )! F (1,K) N (J, m)
where

F (1,K) N (J, m) (t, z 1 ) := T J (h) dT (S d-1 ×R d ) J K -1 dν d V A(T, z 1 , ν, V ) f 0(J K ) N J K (0)) (5.5)
and with (5.6)

A(T, z 1 , ν, V ) := J K -1 i=1 ((v i+1 -v m i (t i )) • ν i+1 ) and ν = {ν 2 , . . . , ν J K } V = {v 2 , . . . , v J K } .
The definition of A requires to compute the whole pseudo-trajectory on the time interval [0, t] starting at z 1 in order to be able to sample the velocities at the different times T = (t 1 , . . . , t J K -1 ). Note that the contributions of the gain and loss terms in the collision operator C k,k+1 are taken into account by the sign of (v

k+1 -v m k (t k )) • ν k+1 .
In the same way, a branching process associated with the Boltzmann hierarchy can be constructed: given an initial particle z 0 1 = (x 0 1 , v 0 1 ) at time t, a collection of collision times T = (t 1 , . . . , t J K -1 ) and labels of the collision particles m = (m 1 , . . . , m J K -1 ) ∈ M J as well as a collection of velocities v 2 , . . . , v J K and deflection angles ν 2 , . . . , ν J K , the (k +1) th particle z 0 k+1 is added at time t k at the position x 0 m k (t k ) of the particle m k and their velocities are adjusted according to the type of the collision

z 0 m k (t k ) = x 0 m k (t k ), v m k (t k ) , z 0 k+1 (t k ) = x 0 m k (t k ), v k+1 if (v k+1 -v m k (t k )) • ν k+1 < 0 z 0 m k (t k ) = x 0 m k (t + k ), v * m k (t + k ) , z 0 k+1 (t k ) = x 0 m k (t + k ), v * k+1 if (v k+1 -v m k (t + k )) • ν k+1 > 0 .
Then, the corresponding pseudo-trajectory Z 0 k+1 evolves according to the backward free flow denoted by Ψ 0 k+1 during the time interval [t k+1 , t k ] until the next particle creation. As the particles are points, no recollision occurs in this branching process. Notice that u → Z 0 k+1 (u) is pointwise left-continuous on [0, t k ]. The counterpart of the integral (5.2) in the series g (1,K) α (t) in (4.12) can be formally rewritten as follows

G (1,K) (J) (t, z 1 ) = T J (h) dT S 0 1 (t -t 1 )C 0 1,2 S 0 2 (t 1 -t 2 )C 0 2,3 . . . S 0 J K (t J K -1 )g 0(J K ) = m∈M J G (1,K) (J, m) (5.7)
where the integral is over the pseudo-trajectories

(5.8) G (1,K) (J, m) (t, z 1 ) := T J (h) dT (S d-1 ×R d ) J K -1 dν d V Â(T, z 1 , ν, V ) g 0(J K ) (Z 0 J K (0)),
with  defined as in (5.6) but with respect to the Boltzmann hierarchy pseudo-trajectories.

t t 1 v 1 (ν 2 , v 2 ) t 2 ε Figure 4
. The first stages of both pseudo-trajectories are depicted up to the occurence of a recollision. The BBGKY pseudo-trajectories are represented with plain arrows, whereas the Boltzmann pseudo-trajectories correspond to the dashed arrows. At time t, the particle with label 1 in the BBGKY hierarchy is a ball of radius ε centered at position x 1 and the particle in the Boltzmann hierarchy is depicted as a point located at x 0 1 = x 1 . At time t 1 the second particle is added and at time t 2 the third. Both hierarchies are coupled, but a small error in the particle positions of order ε can occur at each collision. In this figure, a recollision between the first and the second particle of the BBGKY pseudo-trajectories occurs and after this recollision the Boltzmann and the BBGKY pseudo-trajectories are no longer close to each other. Indeed the BBGKY trajectories are deflected after the recollision, instead the ideal particles do not collide and follow a straight line (see the dashed arrows). Note that before the recollisions the trajectories of z 1 and z 0 1 are identical and therefore the plain and the dashed arrows overlap.

To show that F (1,K) N (J, m) and G (1,K) (J, m) are close to each other when N diverges, we shall prove that the pseudo-trajectories Z and Z 0 can be coupled in order to remain very close to each other up to a small error (see Figure 4)

• due to the micro-translations εν k+1 of the added particle at each collision time t k • excluding the possible recollisions on the interval ]t k , t k-1 [ along the flow S k , which do not occur for the free flow S 0 k .

The proof of the convergence follows the arguments of [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF]. This will be achieved by constructing in (5.18), a set of deflection angles and velocities (B(J, T, m)

) c ⊂ S d-1 × R d J k -1
such that the pseudo-trajectories Z induced by this set have no recollisions and therefore remain very close to the pseudo-trajectories Z 0 associated to the free flow. Furthermore, the measure of B(J, T, m) tends to 0 when N goes to infinity. Finally, in Section 5.3, all the estimates will be combined to derive a quantitative bound on

F (1,K) N (J, m) -G (1,K) (J, m).
5.2. Reduction to non-pathological trajectories.

5.2.1. The elementary step. The set of good configurations with k particles will be such that the particles remain at a distance ε 0 ε for a time t, i.e. that they belong to the set

G k (ε 0 ) := Z k ∈ T dk × R dk ∀u ∈ [0, t], ∀i = j, d(x i -u v i , x j -u v j ) ≥ ε 0
where d denotes the distance on the torus T d . For particles in G k (ε 0 ), the transport Ψ k coincides with the free flow. Fix ā ε 0 . Thus, if at time t the configurations Z k , Z 0 k are such that

(5.9) ∀i ≤ k , |x i -x 0 i | ≤ ā , v i = v 0 i and that Z 0 k belongs to G k (ε 0 ), then the configurations Ψ k (u)Z k , Ψ 0 k (u)Z 0 k will remain at distance less than ā for u ∈ [0, t].
We are going to show that the good configurations are stable by adjunction of a (k + 1) thparticle next to the particle labelled by

m k ≤ k. More precisely, let Z 0 k = (X 0 k , V k ) be in G k (ε 0 ) and Z k = (X k , V k ) with positions close to X 0
k and same velocities (cf. (5.9)). Then, by choosing the velocity v k+1 and the deflection angle ν k+1 of the new particle k + 1 outside a bad set B m k k (Z 0 k ), both configurations Z k and Z 0 k will remain close to each other. Of course, immediately after the adjunction, the particles m k and k + 1 will not be at distance ε 0 , but v k+1 , ν k+1 will be chosen such that the particles drift rapidly far apart and after a short time δ > 0 the configurations Z k+1 and Z 0 k+1 will be again in the good sets G k+1 (ε 0 /2) and G k+1 (ε 0 ). This stability result was obtained in [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] and is stated below. We shall restrict to bounded velocities taking values in the ball B E := v ∈ R d , |v| ≤ E for a given large parameter E > 0 to be tuned later on.

Proposition 5.1 ([21]

). We fix parameters ā, ε 0 , δ such that (5.10)

A K+1 ε ā ε 0 min(δE, 1)
.

Given Z 0 k = (X 0 k , V k ) ∈ G k (ε 0 ) and m k ≤ k, there is a subset B m k k (Z 0 k ) of S d-1 × B E of small measure (5.11) B m k k (Z 0 k ) ≤ Ck E d ā ε 0 d-1 + E d (Et) d ε d-1 0 + E ε 0 δ d-1
such that good configurations close to Z 0 k are stable by adjunction of a collisional particle close to the particle x 0 m k in the following sense. Let Z k = (X k , V k ) be a configuration of k particles satisfying (5.9

), i.e. |X k -X 0 k | ≤ ā. Given (ν k+1 , v k+1 ) ∈ (S d-1 × B E ) \ B m k k (Z 0 k ), a new particle with velocity v k+1 is added at x m k + εν k+1 to Z k and at x 0 m k to Z 0 k . Two possibilities may arise • For a pre-collisional configuration ν k+1 • (v k+1 -v m k ) < 0 then (5.12) ∀u ∈]0, t] , ∀i = j ∈ [1, k] , d(x i -u v i , x j -u v j ) > ε , ∀j ∈ [1, k] , d(x m k + εν k+1 -u v k+1 , x j -u v j ) > ε .
Moreover after the time δ, the k + 1 particles are in a good configuration

(5.13) ∀u ∈ [δ, t] , (X k -uV k , V k , x m k + εν k+1 -u v k+1 , v k+1 ) ∈ G k+1 (ε 0 /2) (X 0 k -uV k , V k , x 0 m k -u v k+1 , v k+1 ) ∈ G k+1 (ε 0 ) .
• For a post-collisional configuration ν k+1 • (v k+1 -v m k ) > 0 then the velocities are updated

(5.14) ∀u ∈]0, t] ,          ∀i = j ∈ [1, k] \ {m k } , d(x i -u v i , x j -u v j ) > ε , ∀j ∈ [1, k] \ {m k } , d(x m k + εν k+1 -u v * k+1 , x j -u v j ) > ε , ∀j ∈ [1, k] \ {m k } , d(x m k -u v * m k , x j -u v j ) > ε , d(x m k -u v * m k , x m k + εν k+1 -u v * k+1 ) > ε .
Moreover after the time δ, the k + 1 particles are in a good configuration

∀u ∈ [δ, t], {x j -u v j , v j } j =m k , x m k -u v * m k , v * m k , x m k + εν k+1 -u v * k+1 , v * k+1 ∈ G k+1 (ε 0 /2), {x 0 j -u v j , v j } j =m k , x 0 m k -u v * m k , v * m k , x 0 m k -u v * k+1 , v * k+1 ∈ G k+1 (ε 0 ) .

5.2.2.

Induction procedure for the pseudo-trajectories. Using the elementary step of Section 5.2.1, we are going to construct in Proposition 5.3 a coupling between the BBGKY and Boltzmann pseudo-trajectories, defined in Section 5.1, such that both trajectories remain close for all times up to a small error. In particular, this proof shows that recollisions may occur for the BBGKY pseudo-trajectories only for a set of configurations at time 0 in D J K ε with small measure.

As the stability of the good configurations (proved in Proposition 5.1) requires a delay δ > 0 in between 2 collisions, we introduce a modified set of collision times

T J,δ (h) := T = (t 1 , . . . , t J K -1 ) / t i < t i-1 -δ , (t J k , . . . , t J k-1 +1 ) ∈ [t -kh, t -(k -1)h] .
(5.16)

The following statement is analogous to Lemma 14.1.1 of [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF].

Proposition 5.3. Fix J = (j 1 , . . . , j K ), m = (m 1 , . . . , m J K -1 ) ∈ M J and T ∈ T J,δ (h). Let the pseudo-trajectories Z i = (X i , V i ), Z 0 i = (X 0 i , V i
) be defined inductively by choosing at each collision time t i a deflection angle ν i+1 and a velocity v i+1 such that

(ν i+1 , v i+1 ) ∈ S d-1 × B E \ B m i i (Z 0 i (t i )) and i+1 k=1 v 2 k < E 2 .
The velocities of both pseudo-trajectories coincide as well as the positions x 1 (u) = x 0 1 (u) for u ∈ [0, t]. Furthermore, for ε sufficiently small

(5.17) ∀i ≤ J K -1, ∀ ≤ i + 1, |x (t i+1 ) -x 0 (t i+1 )| ≤ εi .
As a consequence of this proposition, we define a bad set of velocities and deflection angles for the pathological pseudo-trajectories

B(z 1 , J, T, m) := (ν i , v i ) 2≤i≤J K ∈ S d-1 × B E J K -1 J K k=1 v 2 k < E 2 and ∃i 0 ≤ J K -1 such that ∀i < i 0 , (ν i+1 , v i+1 ) ∈ B m i i (Z 0 i (t i )) c (5.18) and (ν i 0 +1 , v i 0 +1 ) ∈ B m i 0 i 0 (Z 0 i 0 (t i 0 )) .
Proof. We proceed by induction on i, the index of the time variables t i for 1 ≤ i ≤ J K -1.

The recursion hypothesis at step i is

(5.19) Z 0 i (t i ) ∈ G i (ε 0 ) and ∀ ≤ i , |x (t i ) -x 0 (t i )| ≤ ε(i -1) , v (t i ) = v 0 (t i ) .
We first notice that by construction, z 1 (t 1 ) = z 0 1 (t 1 ), so (5.19) holds for i = 1. The initial configuration containing only one particle, there is no possible recollision! Assume that (5.19) holds up to some i ≤ J K -1 and let us prove that (5.19) holds for i+1. We shall consider two cases depending on whether the particle adjoined at time t i is pre-collisional or post-collisional.

• Let us start with the case of pre-collisional velocities (v i+1 , v m i (t i )) at time t i . We recall that the particle is adjoined in such a way that (ν i+1 , v i+1 ) belongs to

S d-1 × B E \ B m i i (Z 0 i (t i )). The new configuration Z 0 i+1 satisfies for all u ∈]t i+1 , t i ] ∀ ≤ i , x 0 (u) = x 0 (t i ) + (u -t i )v (t i ) , v (u) = v (t i ) , x 0 i+1 (u) = x 0 m i (t i ) + (u -t i )v i+1 , v i+1 (u) = v i+1 .
Since t i -t i+1 > δ, Proposition 5.1 implies that Z 0 i+1 (t i+1 ) will be in G i+1 (ε 0 ). Now let us study Z i+1 the BBGKY pseudo-trajectory. Provided that ε is sufficiently small, by the induction assumption (5.19) and the fact that A K+1 ε ≤ ā (see (5.10)), we have

∀ ≤ i, |x (t i ) -x 0 (t i )| ≤ ε(i -1) ≤ ā .
Since Z 0 i (t i ) belongs to G i (ε 0 ), Proposition 5.1 implies that backwards in time, there is free flow for Z i+1 . In particular, (5.20)

∀ < i + 1 , x (u) = x (t i ) + (u -t i )v (t i ) , v (u) = v (t i ) , x i+1 (u) = x m i (t i ) + εν i+1 + (u -t i )v i+1 , v i+1 (u) = v i+1 .
Therefore, the velocities of both configurations coincide and by the induction assumption (5.19)

∀ ≤ i + 1 , ∀u ∈]t i+1 , t i ] , |x (u) -x 0 (u)| ≤ ε(i -1) + ε ≤ εi
where we used that in (5.20) there is a shift by at most ε.

• The case of post-collisional velocities (v i+1 , v m i (t i )) at time t i is identical up to a scattering of the velocities v i+1 , v m i in v * i+1 , v * m i . Note that the constraint i+1 k=1 |v 2 k | < E 2 2 implies that both velocities v * i+1 , v * m i remain in B E .
This concludes the proof of Proposition 5.3.

5.3.

Estimate of the error term. We turn now to the main goal of this section and use the coupling of Proposition 5.3 between the hierachies to show that for K log log N and αt (log log N ) (A-1)/A then (

K) N -g (1,K) α L ∞ ([0,t]×T d ×R d ) → 0 5.21) f (1,
with an explicit rate of convergence when N diverges. The coupling of Proposition 5.3 can be implemented only for a reduced set of velocities taking values in B E and for collision times separated at least by δ. Thus the first step will be to estimate the cost of cutting-off the large velocities and the collision time separation in (5.5) and (5.8). Then in Section 5.3.4, the parameters δ, E and K will be tuned and the error term evaluated.

5.3.1. Energy truncation. Given E > 0, define the large velocity cut-off for f

(1,K) N introduced in (4.10) as f (1,K) N,E := J ε (d-1)(J K -1) (N -1)! (N -J K )! m∈M J F (1,K) N,E (J, m)
where J stands for n 1 -1

j 1 =0 • • • n K -1 j K =0
and the velocities in the integral (5.5) are truncated

F (1,K) N,E (J, m) (t, z 1 ) (5.22)
:= T J (h) dT (S d-1 ×B E ) J K -1 dν d V A(T, z 1 , ν, V ) 1 {H J K (Z J K (0))≤ E 2 2 } f 0(J K ) N Z J K (0)
where A was defined in (5.6) and

H k (Z k ) = 1 2 k i=1 |v i | 2 .
In the same way, for g (1,K) α in (4.12), the large velocity cut-off is defined as

g (1,K) α,E := J α J K -1 m∈M J G (1,K) E (J, m)
where the velocities in the integral (5.8) are truncated

G (1,K) E (J, m) (t, z 1 ) (5.23) := T J (h) dT (S d-1 ×B E ) J K -1 dν d V Â(T, z 1 , ν, V ) 1 {H J K (Z 0 J K (0))≤ E 2 2 } g 0(J K ) (Z 0 J K (0)) .
Then, we have the following error estimate.

Proposition 5.4. There is a constant C depending only on β and d such that, as N goes to infinity in the scaling N ε d-1 α -1 ≡ 1, the following bounds hold:

f (1,K) N -f (1,K) N,E L ∞ ([0,t]×T d ×R d ) + g (1,K) α -g (1,K) α,E L ∞ ([0,t]×T d ×R d ) ≤ A K(K+1) (Cαt) A K+1 e -β 4 E 2 ρ 0 L ∞ , with A, K as in Proposition 4.3.
Proof. We first consider the BBGKY hierarchy. Since the kinetic energy is preserved by the transport S k , the difference (f

(1,K) N -f (1,K)
N,E ) can be bounded from above by estimating the contribution of the pseudo-trajectories such that

{H J K (Z J K (0)) ≥ E 2
2 } at time 0. Note that from (4.6) (5.24)

1 {H J K (Z J K )≥ E 2 2 } f 0(J K ) N ε,J K ,β/2 ≤ f 0(J K ) N ε,J K ,β e -β 4 E 2 ≤ C J K e -β 4 E 2 ρ 0 L ∞ .
By Lemma 4.2, we get

F (1,K) N (J, m) -F (1,K) N,E (J, m) L ∞ ([0,t]×T d ×R d ) ≤ |Q| 1,J K (t)1 {H J K (Z J K )≥ E 2 2 } f 0(J K ) N L ∞ ([0,t]×T d ×R d ) ≤ Ct (β/2) (d+1)/2 J K -1 1 {H J K (Z J K )≥ E 2 2 } f 0(J K ) N ε,J K ,β/2 .

It follows that

F (1,K) N (J, m) -F (1,K) N,E (J, m) L ∞ ([0,t]×T d ×R d ) ≤ (Ct) A K+1 e -β 4 E 2 ρ 0 L ∞
thanks to (5.24) and to the fact that J K ≤ N K ≤ A K+1 . A similar estimate holds for the Boltzmann hierarchy. Summing over all possible choices of j k proves the proposition, recalling that in the Boltzmann-Grad scaling

(ε d-1 ) J K -1 (N -1)! (N -J K )! ≤ α J K -1 .
Proposition 5.4 is proved.

5.3.2. Time separation. We choose a small parameter δ > 0 such that A K δ h and estimate the error for separating the collision times by at least δ. The time cut-off of the pseudotrajectories is defined as

f (1,K) N,E,δ := J ε (d-1)(J K -1) (N -1)! (N -J K )! m∈M J F (1,K) N,E,δ (J, m) (5.25)
where the time integrals are restricted to the set T J,δ (h) defined in (5.16)

F (1,K) N,E,δ (J, m) (t, z 1 ) := T J,δ (h) dT (S d-1 ×B E ) J K -1 dν d V A(T, z 1 , ν, V ) 1 {H J K (Z J K (0))≤ E 2 2 } f 0(J K ) N (Z J K (0))
with A(t, z 1 , ν, V ) as in (5.6). In the same way, for the Boltzmann hierarchy, we set

g (1,K) α,E,δ := J α J K -1 m∈M J G (1,K) E,δ (J, m)
where the separation time cut-off is defined as

G (1,K) E,δ (J, m) (t, z 1 ) := T J,δ (h)
dT

(S d-1 ×B E ) J K -1 dν d V Â(T, z 1 , ν, V ) 1 {H J K (Z 0 J K (0))≤ E 2 2 } g 0(J K ) (Z 0 J K (0)).
Then the following holds.

Proposition 5.5. There is a constant C depending only on β and d such that, as N goes to infinity in the scaling N ε d-1 α -1 ≡ 1, the following holds

f (1,K) N,E -f (1,K) N,E,δ L ∞ ([0,t]×T d ×R d ) + g (1,K) α,E -g (1,K) α,E,δ L ∞ ([0,t]×T d ×R d ) ≤ A (K+2)(K+1) (Cαt) A K+1 δ t ρ 0 L ∞ , (5.26)
with A, K as in Proposition 4.3.

Proof. Given J, m the difference (F

(1,K) N,E -F (1,K)
N,E,δ )(J, m) involves the integration over two consecutive times such that |t i+1 -t i | ≤ δ. This leads to a contribution δt J K -2 /(J K -2)! instead of t J K -1 /(J K -1)! and there are J K -1 possible choices for the collision with a short time separation. Modifying accordingly the estimates of Lemma 4.2, we get for a given J

m∈M J F (1,K) N,E -F (1,K) N,E,δ (J, m) L ∞ ([0,t]×T d ×R d ) ≤ (Cαt) A K+1 (A K+1 ) 2 δ t ρ 0 L ∞ ,
where we used that J K ≤ A K+1 . Summing over all possible choices of j k leads to an extra factor A K(K+1) as in (5.26).

A similar estimate holds in the Boltzmann case and completes the proof.

5.3.3.

Neglecting the pathological pseudo-trajectories. We now reduce the domain of integration of the velocities and deflection angles outside the set B(J, T, m) defined in (5.18) in order to remove the pathological pseudo-trajectories. We set

f (1,K) N,E,δ = J α J K -1 m∈M J ε d-1 α J K -1 (N -1)! (N -J K )! F (1,K) N,E,δ (J, m) (5.27) where F (1,K) N,E,δ (J, m) (t, z 1 ) (5.28) := T J,δ (h) dT B(J,T,m) c dν d V A(T, z 1 , ν, V ) 1 {H J K (Z J K (0))≤ E 2 2 } f 0(J K ) N (Z J K (0))
with A(t, z 1 , ν, V ) as in (5.6). In the same way, we define

g(1,K) α,E,δ = J α J K -1 m∈M J G(1,K) E,δ (J, m) (5.29)
where the domain of integration is restricted to the complement of B(J, T, m)

G(1,K) E,δ (J, m) (t, z 1 ) (5.30) := T J,δ (h) dT B(J,T,m) c dν d V Â(T, z 1 , ν, V ) 1 {H J K (Z 0 J K (0))≤ E 2
As a consequence of Proposition 5.1 and of the continuity estimates in Lemma 4.2, the error induced by neglecting the pathological pseudo-trajectories can be estimated from above. Proposition 5.6. Let ā, ε 0 , δ satisfying (5.10). There is a constant C depending only on β and d such that, as N goes to infinity in the scaling N ε d-1 α -1 ≡ 1, the following holds

g (1,K) α,E,δ -g (1,K) α,E,δ L ∞ ([0,t]×T d ×R d ) + f (1,K) N,E,δ -f (1,K) N,E,δ L ∞ ([0,t]×T d ×R d ) ≤ A (K+2)(K+1) (Cαt) A K+1 E d ā ε 0 d-1 + E d (Et) d ε d-1 0 + E ε 0 δ d-1 ρ 0 L ∞ .
Proof. The proof follows the same lines as the proofs of Propositions 5.4 and 5.5. In the usual continuity estimate for the elementary collision operator, the integration with respect to velocity brings a factor (2π/β) d/2 , while removing the integration over the pathological set B m k k gives an error (5.31)

Ck E d ā ε 0 d-1 + E d Et d ε d-1 0 + E ε 0 δ d-1
according to Proposition 5.1. For a given J, there are J K -1 ≤ A K+1 possible choices of the integral to be modified. Therefore, the estimate on the collision operator leads to

m∈M J F (1,K) N,E,δ -F (1,K) N,E,δ (J, m) L ∞ ([0,t]×T d ×R d ) ≤ (Ct) A K+1 A 2(K+1) E d ā ε 0 d-1 + E d (Et) d ε d-1 0 + E ε 0 δ d-1 ρ 0 L ∞
where as previously C depends only on d and β. The term A 2(K+1) comes from the A K+1 possible choices of the integral to be modified and from the additional factor k ≤ A K+1 in (5.31). Finally summing over all the possible choices of J = (j 1 , . . . , j K ) provides the additional factor A K(K+1) in the estimate. Similar bounds hold also for the Boltzmann hierarchy. This completes the Proposition.

Once the pathological pseudo-trajectories have been removed, the integrals (5.28) and (5.30) differ only by the small error on the positions Z J K (0), Z 0 J K (0) and by the initial data f 0(J K ) N and g 0(J K ) . Thus, one gets Proposition 5.7. There is a constant C depending only on β and d such that, as N goes to infinity in the scaling N ε d-1 = α, the following holds

f (1,K) N,E,δ - g(1,K) α,E,δ L ∞ ([0,t]×T d ×R d ) ≤ A K(K+1) (Cαt) A K+1 A 2(K+1) N + αε ρ 0 L ∞ .
Proof. There are 2 sources of discrepancies between (5.27) and (5.29).

• The prefactors in the collision operators : In (5.27), the elementary collision operators have prefactors of the type (N -k)ε d-1 /α that can be replaced in the limit by 1. For fixed J K , the corresponding error is

1 - (N -1) . . . (N -J K + 1) N J K +1 ≤ C J 2 K N
which, combined with the bound on the collision operators, leads to an error of the form

A K(K+1) (Cαt) A K+1 A 2(K+1) N ρ 0 L ∞ • • Discrepancy between f 0(J K ) N (Z J K (0)
) and g 0(J K ) (Z 0 J K (0)) : First of all, we note that for the coupled pseudo-trajectories g 0(J K ) (Z J K (0)) = g 0(J K ) (Z 0 J K (0)). Indeed, by construction both pseudo-trajectories have the same velocities and x 1 = x 0 1 . The differences between the two configurations are only on the positions of the particles added and g 0(J K ) is independent of these positions. By Proposition 5.3, the initial data satisfies Z J K (0) ∈ G J K (ε 0 /2). According to Proposition 3.3, we have

1 G J K (ε 0 /2) (f 0(J K ) N -g 0(J K ) ) 0,J K ,β ≤ ρ 0 L ∞ C J K αε .
Using the continuity estimate in Lemma 4.2, we then deduce that the error due to the initial data can be controlled by ρ 0 L ∞ A K(K+1) (Cαt) A K+1 αε. This concludes Proposition 5.7.

f (1,K) N -g (1,K) α L ∞ ([0,t]×T d ×R d ) ≤ ρ 0 L ∞ ε d-1 d+1 exp C (log N ) 1/2 log log N .
In particular, Estimate (5.21) follows from Proposition 5.8.

Proof. We write (5.10). By gathering together the estimates in Propositions 5.4, 5.5, 5.6 and 5.7, we see that there exists C depending only on β and d such that, as N goes to infinity in the scaling N ε d-1 α -1 ≡ 1, the following holds

f (1,K) N -g (1,K) α L ∞ ≤ f (1,K) N - f (1,K) N,E,δ L ∞ + g (1,K) α - g(1,K) α,E,δ L ∞ + f (1,K) N,E,δ - g(1,K) α,E,δ L ∞ . Let ā, ε 0 , δ, E satisfying
f (1,K) N -g (1,K) α L ∞ ([0,t]×T d ×R d ) ≤ A K(K+1) (Cαt) A K+1 e -β 4 E 2 + A 2(K+1) δ t ρ 0 L ∞ + A (K+2)(K+1) (Cαt) A K+1 E d ā ε 0 d-1 + E d Et d ε d-1 0 + E ε 0 δ d-1 ρ 0 L ∞ + A K(K+1) (Cαt) A K+1 A 2(K+1) N + εα ρ 0 L ∞ with A, K introduced in Proposition 4.3.
To derive the upper bound (5.33), we choose for the parameters the following orders of magnitude δ ∼ ε (5.33) can be deduced in the scaling (5.32)

d-1 d+1 , ε 0 ∼ ε d d+1 , E ∼ | log ε|, ā = A K+1 ε . This leads to f (1,K) N -g (1,K) α L ∞ ([0,t]×T d ×R d ) ≤ (C αt) A K+1 A 2K(K+1) ε d-1 d+1 | log ε| d + ε d-1 ρ 0 ∞ from which
since A K ≤ √ log N .
Equipped with all these estimates, we prove now Theorem 2.2.

Proof of Theorem 2.2. Propositions 4.3 and 5.8 imply that with the scaling (5.32)

f (1) N -g α L ∞ ([0,t]×T d ×R d ) ≤ C γ A + C 0 ε d-1 d+1 exp C (log N ) 1/2 log log N ρ 0 L ∞ ≤ C (αt) A/(A-1) log log N A ρ 0 L ∞ ,
where we have used the relation γ = (αt) A/(A-1)

CK of (4.13) with the choice K = log log N 2 log A . This conludes the proof of Theorem 2.2. Note that the relevant scaling for this upper bound is αt = o (log log N ) (A-1)/A . 6. Proof of the diffusive limit : proof of Theorem 2.3

In Theorem 2.2, we have shown that the tagged particle distribution f

N (t, x, v) remains close to M β (v)ϕ α (t, x, v)
where ϕ α solves the linear-Boltzmann equation (1.3) on T d × R d , with initial data ρ 0 (x). More generally, our proof implies that the whole trajectory of the tagged particle {x 1 (s)} s≤T can be approximated with high probability by the trajectory of {x 0 1 (s)} s≤T (see Lemma 5.3). The latter process is much simpler to study as its velocities are given by a Markov process. These two points of view lead to two strategies to prove the diffusive limit. We first present an analytic approach to show that ϕ α (ατ, x, v) can be approximated by the diffusion (2.11). Then we turn to an alternative method to show the convergence of the trajectory to a brownian motion which will rely on probabilistic estimates for {x 0 1 (ατ )} τ ≤T . In the following the macroscopic time variable will be denoted by τ ∈ [0, T ].

6.1. Convergence to the heat equation. In this section we prove the result (2.12) stating the convergence of f N (ατ, x, v) to M β (v)ρ(τ, x) where ρ solves the heat equation (2.11) on T d , with initial data ρ 0 (x). We show in Paragraph 6.1.1 that this can be reduced to proving that ϕ α (ατ, x, v) can be approximated by a diffusion, which is a standard procedure (see [START_REF] Bardos | Diffusion approximation and computation of the critical size[END_REF]). For the sake of completeness, we recall the salient features of the proof in Paragraphs 6.1.2 and 6.1.3. 6.1.1. Approximation by the linear Boltzmann equation. The explicit convergence rate provided in Theorem 2.2 implies in particular that for any τ > 0 and any α > 1, in the limit N → ∞, N ε d-1 α -1 = 1, one has (6.1) f

N (ατ, x, v) -M β (v)ϕ α (ατ, x, v) L ∞ (T d ×R d ) ≤ C α 2 τ (log log N ) A-1 A A 2 A-1 (1)
, where A ≥ 2 can be taken arbitrarily large. It is therefore possible to take the limit α → ∞ while conserving a small right-hand side in (6.1), as soon as α (log log N )

A-1

2A

.

Let us define ϕ α (τ, x, v) := ϕ α (ατ, x, v) , which satisfies (6.2)

∂ τ ϕ α + α v • ∇ x ϕ α + α 2 L ϕ α = 0 .
Then (2.12) follows directly from the following result

(6.3) sup τ ∈[0,T ] sup (x,v)∈T d ×R d M β (v) ϕ α (τ, x, v) -ρ(τ, x) → 0 in the limit α → ∞.
The rest of this paragraph is devoted to the proof of (6.3). Notice that by the maximum principle on the heat equation, we may assume without loss of generality (up to regularizing ρ 0 ) that ρ 0 belongs to C 4 (T d ), which will be useful at the end of the proof.

6.1.2. Hilbert's expansion. The formal Hilbert expansion consists in writing an asymptotic expansion of ϕ α in terms of powers of α -1

ϕ α (τ, x, v) = ρ 0 (τ, x, v) + 1 α ρ 1 (τ, x, v) + 1 α 2 ρ 2 (τ, x, v) + .
. . , in plugging that expansion in Equation (6.2), and in canceling successively all the powers of α. This gives formally the following set of equations (where we have considered only the O(1), O(α) and O(α 2 ) terms) (6.4)

L ρ 0 = 0 , v • ∇ x ρ 0 + L ρ 1 = 0 , ∂ τ ρ 0 + v • ∇ x ρ 1 + L ρ 2 = 0 .
In order to find the expressions for ρ 1 and ρ 2 , as well as the equation on ρ 0 (which we expect to be the heat equation), it is necessary to be able to invert the operator L. This is made possible by the following result, whose proof can be found in [START_REF] Hilbert | Begründung der kinetischen Gastheorie[END_REF] (in the case of the linearized Boltzmann equation, but it can easily be adapted to our situation). In the following, we define

a β (v) := S d-1 ×R d M β (v 1 ) (v -v 1 ) • ν + dνdv 1 .
The proof of the next result consists in noticing the decomposition L = a β (v) Id-K, where Id stands for the identity and K is a compact operator. Lemma 6.1. The operator L is a Fredholm operator of domain L 2 (R d , a β M β dv) and its kernel reduces to the constant functions. In particular, L is invertible on the set of functions

g ∈ L 2 (R d , a β M β dv), R d g(v) M β (v)dv = 0 .
Note that the first equation in (6.4) therefore reflects the fact that ρ 0 does not depend on v.

We define the vector b

(v) = b k (v) k≤d with R d b(v) M β (v)dv = 0, by (6.5) Lb(v) = v.
Returning to (6.4), we have

ρ 1 (τ, x, v) = ρ 1 (τ, x, v) + ρ 1 (τ, x) , with ρ 1 (τ, x, v) := -b(v) • ∇ x ρ 0 (τ, x) and ρ 1 ∈ Ker L .
Next we consider the last equation in (6.4) and we notice that for ρ 2 to exist it is necessary for ∂ τ ρ 0 + v • ∇ x ρ 1 to belong to the range of L. Since ρ 0 does not depend on v, this means that (6.6)

∂ τ ρ 0 + R d v • ∇ x ρ 1 (τ, x, v)M β (v) dv = 0 .
We then define the diffusion matrix D(v) = D k, (v) k, ≤d , again with

R d D k, (v) M β (v)dv = 0, by (6.7) LD(v) := v ⊗ b(v) - R d v ⊗ b(v) M β (v)dv .
From the symmetry of the model, one can check (see [START_REF] Desvillettes | A remark concerning the Chapman-Enskog asymptotics[END_REF] for instance) that there is a function γ such that b(v) = γ(|v|)v .

Then an easy computation shows that ρ 0 = ρ where

∂ τ ρ -κ β ∆ x ρ = 0 ,
while the diffusion coefficient is given by (6.8)

κ β := 1 d R d vL -1 v M β (v)dv = 1 d R d γ(|v|)|v| 2 M β (v)dv ,
and where we used the symmetry of b to derive the last equality. Finally we have

ρ 2 (τ, x, v) = ρ 2 (τ, x, v) + ρ 2 (τ, x) -b(v) • ∇ x ρ 1 (τ, x) ,
∂ τ Ψ α + α v • ∇ x Ψ α + α 2 LΨ α = S α ,
where the error term S α is given by (6.10)

S α (τ, x, v) := 1 α ∂ τ ρ 1 (τ, x, v) + v • ∇ x ρ 2 (τ, y, v) + 1 α ∂ τ ρ 2 (τ, y, v) . Defining R α (τ, x, v) := Ψ α (τ, x, v) -ϕ α (τ, x, v)
we have thanks to (6.2)

∂ τ R α + α v • ∇ x R α + α 2 LR α = S α
and the result (6.3) then follows from the maximum principle which states that

M β R α L ∞ ([0,T ]×T d ×R d ) ≤ C(T ) M β R α (0) L ∞ (T d ×R d ) + M β S α L ∞ ([0,T ]×T d ×R d ) .
We note that S α involves spatial derivatives of ρ of order at most 4, thus from the maximum principle for the heat equation, each term of M β S α is bounded in L ∞ norm by α -1 . The same clearly holds for the initial data

M β R α (0, x, v) since R α (0, x, v) = Ψ α (0, x, v) -ϕ α (0, x, v) = 1 α ρ 1 (0, x, v) + 1 α 2 ρ 2 (0, x, v) .
It follows that

M β (Ψ α -ϕ α ) L ∞ ([0,T ]×T d ×R d ) ≤ C(T )
α and thanks to (6.9), the convergence result (6.3) is proved. Remark 6.2. We have considered here the case when ρ 0 = ρ 0 (x). In the case of ill-prepared initial data, namely if ρ 0 = ϕ 0 (x, v), then the same analysis works provided the following ansatz is used

Ψ α (τ, x, v) := ρ(τ, x) + 1 α ρ 1 (τ, x, v) + 1 α 2 ρ 2 (τ, x, v) + Π ⊥ e -ατ L ϕ 0 ,
where Π ⊥ is the orthogonal projector onto (Ker L) ⊥ .

6.2. Convergence to the brownian motion. Let us denote the tagged particle by Ξ(τ ) := x 1 (ατ ) .

In the following, E N , P N will refer to its expectation and probability with respect to the initial data sampled from the density f 0 N . To prove the convergence of the tagged particle to a brownian motion, one needs to check (see [START_REF] Billingsley | Probability and measure[END_REF], Chapter 2)

• the convergence of the marginals of the tagged particle sampled at different times (6.11) lim

N →∞ E N h 1 Ξ(τ 1 ) . . . h Ξ(τ ) = E h 1 B(τ 1 ) . . . h B(τ ) ,
where {h 1 , . . . , h } is a collection of continuous functions in T d . Notice that these marginals refer to time averages and not to the number of particles. • the tightness of the sequence, that is for any τ ∈ [0, T ] (6.12) ∀ξ > 0, lim

η→0 lim N →∞ P N sup τ <σ<τ +η Ξ(σ) -Ξ(τ ) ≥ ξ = 0 .
Note that (6.11) requires to understand time correlations and thus we are going to adapt Theorem 2.2 to this new framework.

Step 1. Finite dimensional marginals. First, we are going to rewrite the time correlations in terms of collision trees. A similar approach was devised in Lebowitz, Spohn [START_REF] Lebowitz | Steady state self-diffusion at low density[END_REF] to derive an information on the true particle trajectories (in the physical space) from the Duhamel series. Let t 1 < • • • < t be an increasing collection of times and H = {h 1 , . . . , h } a collection of smooth functions. Define the biased distribution at time t > t as follows

T N d ×R N d dZ N f N,H (t, Z N )Φ(Z N ) := E N h 1 x 1 (t 1 ) . . . h x 1 (t ) Φ Z N (t) (6.13) = T N d ×R N d dZ N f 0 N (Z N ) h 1 x 1 (t 1 ) . . . h x 1 (t ) Φ Z N (t) ,
for any test function Φ. We stress that by construction the biased distribution f N,H (t, Z N )

• is in general no longer normalized by 1

• is symmetric with respect to the N -1 last variables.

(s) N,H (t) = N -s m=0 Q s,s+m (t -t )f (s+m) N,H (t ) . By construction f N,H (t , Z N ) = f N,H -1 (t , Z N )h (z 1 )
, where the new distribution is now modified by the the first -1 functions. This procedure can be iterated up to the initial time. The backward dynamics can be understood in terms of collision trees which are now weighted by the factor h 1 x 1 (t 1 ) . . . h x 1 (t ) associated with the motion of the tagged particle (6.16)

f (1) N,H (t) = N -1 m 1 +•••+m =0 Q 1,1+m 1 (t -t ) h Q 1+m 1 ,1+m 2 (t -t -1 ) h -1 . . . Q 1+m 1 +•••+m -1 ,1+m 1 +•••+m (t 1 ) f (1+m 1 +•••+m ) N (0) .
This identity holds for any N and any time.

In order to check (6.12), we need also to generalize the identity to consider correlations of the form (6.17) E N h x 1 (t 1 ) -x 1 (s) . . . h x 1 (t ) -x 1 (s)

for a smooth function h with s < t 1 < • • • < t . Using a partition of unity {Γ ξ i } centered at points γ i ∈ T d with mesh ξ, one can approximate h h(x -y) = i,j h(γ i -γ j )Γ ξ j (x)Γ ξ i (y) + O(ξ) .

This allows us to use the identity (6.16) for any accuracy ξ > 0 of the approximation. Thus (6.17) can be computed in terms of collision trees which are now weighted by the factor h x 1 (t 1 ) -x 1 (s) . . . h x 1 (t ) -x 1 (s) .

Step 2. The limit process.

In the Boltzmann Grad limit, the memory of the system is lost and the tagged particle behavior becomes equivalent to a Markov process. We define (6. M β (v i ) . is the backbone of the tree with branchings at each new collision. There cannot be further branches as any collision with a new particle would lead to a cancellation in the collision operator. Thus the trees involving the branches in dashed line do not contribute to the Duhamel series in the Boltzmann hierarchy.

As in (6.16), the distribution can be rewritten in terms of a Duhamel series (6.20)

g α,H (t) = ∞ m 1 +•••+m =0 Q 0 1,1+m 1 (t -t ) h Q 0 1+m 1 ,1+m 2 (t -t -1 ) h -1 . . . Q 0 1+m 1 +•••+m -1 ,1+m 1 +•••+m (t 1 ) g (1+m 1 +•••+m ) α,H (0)
.

This representation allows us to rephrase the Markov chain expectations in terms of the Boltzmann hierarchy. In this series, a lot of cancelations occur (see figure 5). Indeed, the only relevant collision trees are made of a single backbone (the trajectory of z 0 1 ) with branches representing the collisions of z 0 1 with the ideal gas, but no further ramification. In step 3, we shall not use these cancellations and simply compare the series (6. [START_REF] Desvillettes | The linear Boltzmann equation for long-range forces: a derivation from particles[END_REF]) and (6.20) term by term in order to show that in the Boltzmann Grad limit when α √ log log N (6.21) lim α→∞ E M β h 1 x1 (ατ 1 ) . . . h x1 (ατ ) -E N h 1 x 1 (ατ 1 ) . . . h x 1 (ατ ) = 0.

As L has a spectral gap, the invariance principle holds for the position of the Markov process x1 (see [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] Theorem 2.32 page 74). This implies the convergence of the rescaled finite dimensional marginals towards the ones of the brownian motion B with variance κ β (see (6.8)), i.e. that for any smooth functions {h 1 , . . . , h } defined in T d , (6.22) lim α→∞ E M β h 1 x1 (ατ 1 ) . . . h x1 (ατ ) = E h 1 B(τ 1 ) . . . h B(τ ) .

The diffusion coefficient κ β defined in (6.8) can be interpreted in terms of the variance of the position properly rescaled in time (see [START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] page 47).

Step 3. Approximation of the finite dimensional marginals.

We turn now to the proof of (6.21) which combined with (6.22) will show the convergence (6.11) of the marginals of the tagged particle sampled at different times.

Suppose now that the collection H satisfies the uniform bounds on T d (6.23) ∀i ≤ , 0 ≤ h i (x 1 ) ≤ m.

Thus the f (s)

N,H satisfy the maximum principle (4.2) with an extra factor m . The pruning procedure on the collision trees therefore applies also in this case and enables us to restrict to trees with at most A K+1 collisions during the time interval [0, t]. Furthermore, the comparison of the trajectories for f N,H and g α,H can be achieved in the same way as before on a tree with less than A K+1 collisions and no recollisions. Analogous bounds as in Proposition 5.7 can be obtained, but one has to take into account that the trees are now weighted by h 1 x 0 1 (ατ 1 ) . . . h x 0 1 (ατ ) . We recall that the pseudo-trajectories of x 1 and x 0 1 coincide at any time, thus bounds similar to (2.9) hold (6.24) f

•

This implies (6.21), hence (6.11) thanks to (6.22).

Step 4. Tightness.

In order to evaluate (6.12), it is enough to sample the trajectory of the tagged particle at the times τ i = {τ + u N α i} i≤ N for u N = 1 log N and with N := αη/u N . Indeed, we can decompose the path deviations into two terms We shall first evaluate the last term in the right-hand side which involves only events occurring in a microscopic time scale of length u N . Given i ≤ N , let t i := iu N + ατ and t i+1 := u N + t i then

P N sup τ i <σ<τ i+1
Ξ(σ) -Ξ(τ i ) ≥ ξ = P N sup t i <s<t i+1

x 1 (s) -x 1 (t i ) ≥ ξ .

In order to control the tagged particle fluctuations, it is enough to bound its velocity in the time interval [t i , t i+1 ] P N sup

t i <s<t i+1 s t i v 1 (s )ds ≥ ξ ≤ P N t i+1 t i v 1 (s ) ds ≥ ξ ≤ ρ 0 ∞ P N u N 0 v 1 (s ) ds ≥ ξ ,
where we used the maximum principle in the last inequality and P N denotes the dynamics starting from the invariant measure M N,β . Following the strategy in [START_REF] Alexander | Time evolution for infinitely many hard spheres[END_REF] to bound this probability, we write

P N u N 0 v 1 (s ) ds ≥ ξ ≤ exp - ξ u N E N exp 1 u N u N 0 v 1 (s ) ds .
Using Jensen's inequality and the invariant measure, we get

E N exp 1 u N u N 0 v 1 (s ) ds ≤ 1 u N u N 0 ds E N exp v 1 (s ) = E M N,β exp v 1 ≤ c β ,
where c β is a constant depending only on β. Since u N = 1 log N , we have shown that for any ξ > 0, the probability of a deviation in a very short time vanishes when N goes to By comparison, we are going to show that the same result holds also for the tagged particle x 1 . Using (6.25), this will complete the proof of (6.12). Let h ξ (w) = 1 {|w|≤ξ/2} , then it is enough to show that (6.26)

E N N i=1 h ξ x 1 (ατ i ) -x 1 (ατ ) -E M β N i=1 h ξ x1 (ατ i ) -x1 (ατ ) ≤ C ρ 0 ∞ τ α 2
(log log N )

A-1 A

A 2 A-1

•

At this stage, it is enough to use the fact that probabilities of the form (6.17) can also be evaluated in terms of weighted trees as in Step 1. Since h ξ is bounded by 1, the maximum principle applies uniformly in N . The tree decomposition and the reduction to non pathological trajectories hold as in the previous proof. For good pseudo-trajectories, the paths of x 1 and x 0 1 coincide, therefore modifying the Duhamel series by N i=1 h ξ x 1 (ατ i ) -x 1 (ατ ) does not alter the comparison established in Proposition 5.7. This concludes the proof of tightness.

Proposition 4 . 1 .

 41 For any fixed N , denote by f N the solution to the Liouville equation (2.3) with initial data (2.8), and by f (s) N its marginal of order s (4.1)

4. 4 .

 4 Estimates of the remainders. Since we expect the particles to undergo on average one collision per unit of time, the growth of collision trees is typically exponential. Pathological trees are therefore those with super exponential growth. There are two natural ways of defining such pathological trees • either by choosing some fixed h (given for instance by Lanford's proof) and log n k k; • or by fixing n k = A k and letting the elementary time interval h → 0.

  α are close to each other.

5. 3 . 4 .

 34 Estimate of the main term. Finally combining the previous estimates, we get Proposition 5.8. For parameters satisfying (5.1) and such that

with ρ 2 6 . 1 . 3 .

 2613 (τ, x, v) := D(v) : Hess ρ(τ, x) and ρ 2 ∈ Ker L . Proof of the convergence. Now let us prove(6.3). With the notation introduced in the previous paragraph, let us define (6.9) Ψ α (τ, x, v) := ρ(τ, x) + 1 α ρ 1 (τ, x, v) + 1 α 2 ρ 2 (τ, x, v) . Then Ψ α is almost a solution of (6.2): by construction one has

t z 1 Figure 5 .

 15 Figure 5. A collision tree for the Boltzmann hierarchy is depicted. The path of z 1

( 1 )∞ τ α 2 (

 12 N,H (ατ, y, v) -g α,H (ατ, y, v) L ∞ (T d ×R d ) ≤ Cm ρ 0 log log N )

  P N sup τ <σ<τ +η Ξ(σ) -Ξ(τ ) ≥ 2ξ ≤1 -P N N i=1 Ξ(τ i ) -Ξ(τ ) < ξ (6.25) + N i=1 P N sup τ i <σ<τ i+1 Ξ(σ) -Ξ(τ i ) ≥ ξ .

  i+1 Ξ(σ) -Ξ(τ i ) ≥ ξ ≤ c β N exp (-ξ log N ) ----→ N →∞ 0.The tightness for the process x1 derived in[START_REF] Komorowski | Fluctuations in Markov processes. Time symmetry and martingale approximation[END_REF] (Theorem 2.32 page 74) implies that for any ξ > 0 and N = αη/u N i ) -x1 (ατ ) < ξ/2 = 1 .

  By construction f N,H satisfies the Liouville equation for t > t and the marginals f

			(s) N,H obey
	the BBGKY hierarchy (3.1) for s < N . Applying the iterated Duhamel formula (3.5), we get
	(6.15)		f
	The corresponding marginals are
	(6.14)	f	(s)

N,H (t, Z s ) := f N,H (t, Z N ) dz s+1 . . . dz N .

  ds as an additive functional of the Markov chain {v 1 (s)} s≥0 with generator αL introduced in (1.3). Initially (x 1 (0), v1 (0)) is distributed according to ρ 0 (x)M β (v). The expectation associated to this Markov chain is denoted by E M β .Let t 1 < • • • < t be an increasing collection of times and H = {h 1 , . . . , h } a collection of smooth functions. As in (6.13), we define g α,H (t) as the biased distribution of the Markov chain z1 (t) = x1 (t), v1 (t)∀t ∈ [t k , t k+1 [, T d ×R d g α,H (t, z) Φ(z) dz = E M β h 1 x1 (t 1 ) . . . h x1 (t ) Φ z1 (t) ,with t +1 = ∞. One can consider a measure (cf.(3.11)) including as well the background density of an ideal gas. The marginals of this measure are(6.19) g

		t
	18)	x1 (t) = x1 (0) +	v1 (s)
		0

(s) α,H (t, Z s ) = g α,H (t, z 1 ) s i=2

} g 0(J K ) (Z 0 J K (0)).

Acknowledgments. The authors are grateful to Laurent Desvillettes, François Golse, Herbert Spohn and Balint Toth for fruitful discussions and for suggesting interesting bibliographical references. Finally we extend our thanks to the anonymous referees for suggesting many improvements in the manuscript. The work of I.G. has been supported by the grant ANR-12-BS01-0013-01.

(5.15) Proposition 5.1 is the elementary step for adding a new particle. In Section 5.2.2, we are going to show how this step can be iterated in order to build inductively good pseudo-trajectories Z and Z 0 . Note that after adding a new particle, the velocities remain identical at each time in both configurations, but their positions differ due the exclusion condition in the BBGKY hierarchy which induces a shift of ε at each creation of a new particle (see Figure 4). We refer to [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] for a complete proof of Proposition 5.1 and simply recall that it can be obtained from the following control on free trajectories. Lemma 5.2. Given t > 0, and ā > 0 satisfying A K+1 ε ā ε 0 min(δE, 1), consider two points x 0 1 , x 0 2 in T d such that d(x 0 1 , x 0 2 ) ≥ ε 0 , and a velocity v 1 ∈ B E . Then there exists a subset K(x 0 1 -x 0 2 , ε 0 , ā) of R d with measure bounded by

The proof of this lemma is a simple adaptation of Lemma 12.2.1 in [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF], and is given in Appendix B. Note that this is the only point of the convergence proof which differs in the case of the torus T d from the case of the whole space R d . In the case of the torus, there are indeed no longer dispersion properties so waiting for a sufficiently long time, we expect trajectories to go back ε-close to their initial positions.

Appendix A. Asymptotic control of the exclusion

For the sake of completeness, we recall here the proof of Proposition 3.2. We omit all subscripts β to simplify the presentation.

• First step: asymptotic behaviour of the partition function.

We first prove that in the scaling

where κ d denotes the volume of the unit ball in R d . The first inequality is due to the immediate upper bound Z N ≤ Z N -s . Let us prove the second inequality. We have by definition

1≤i =j≤s+1

By Fubini's equality, we deduce

we deduce the lower bound

where we used s ≤ N and the scaling N ε d-1 ≡ α. This implies by induction

That proves (A.1).

• Second step: convergence of the marginals.

Let us introduce the short-hand notation dZ (s+1,N ) := dz s+1 . . . dz N .

We compute for s ≤ N M (s)

We deduce, by symmetry,

From there, the difference

N decomposes as a sum

By (A.1), there holds 1 -Z -1 N Z N -s → 0 as N → ∞, for fixed s. Since M ⊗s is uniformly bounded, this implies that the first term in the right-hand side of (A.3) tends to 0 as N goes to ∞. Besides, by

Given 1 ≤ i ≤ s, there holds by symmetry and Fubini's equality,

By (A.1), we obtain

and the upper bound tends to 0 as N → ∞, for fixed s. This implies convergence to 0 of the second term in the right-hand side of (A.3). This completes the proof of Proposition 3.2.

Appendix B. Recollisions in the torus

We show here how to adapt the arguments of [START_REF] Gallagher | From Newton to Boltzmann : the case of hard-spheres and short-range potentials[END_REF] to prove Lemma 5.2.

• To build the set of "bad velocities", we use the correspondence between the torus and the whole space with periodic structure. Asking that there exists u ∈ [0, t] such that

Then, by the triangular inequality and provided that ε < ā,

In other words, v 1 -v 2 has to belong to a finite union of cones of vertex 0 • at most one of which is of solid angle (ā/ε 0 ) d-1 ;

• the other ones (at most (4Et) d ) are of solid angle c ād-1 . The intersection K(x 1 -x2 , ε 0 , ā) of these cones and of the sphere of radius 2E is of size

• In order to prove the second estimate, we need to refine a little bit the previous argument.

Asking that there exists u ∈ [δ, t] such that

) implies that v 1 -v 2 belongs to the intersection of B 2E (0) and some cone of vertex 0 and solid angle ε d-1 0 . • If |x 0 1 -x 0 2 + k| ≤ 1/4 (which can happen only for one value of k), denoting by n any unit vector normal to x1 -x2 + k, we deduce from (B.1) that u|(v 1 -v 2 ) • n| ≤ 3ε 0 from which we deduce that v 1 -v 2 belongs to the intersection of B 2E (0) and some cylinder of radius ε 0 /δ. The union K δ (x 0 1 -x 0 2 , ε 0 , ā) of these "bad" sets is therefore of size

.

The lemma is proved.