The Brownian Motion as the Limit of a Deterministic System of Hard-spheres

We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit α = Nεd−1 → ∞ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees. 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 Rd, with positions and velocities (xi, vi)1≤i≤N ∈ (D × Rd)N , the dynamics of which is given by (1.1) dxi dt = vi , m dvi dt = − ε ∑ j 6=i ∇Φ (xi − xj 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 Nεd = 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 [36, 20, 38] 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 Nεd 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 ≡ Nεd−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. Date: February 25, 2015. 1 2 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND 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). Microscopic descrip+on System of N par.cles of size ε Newton’s equa.ons Mesoscopic descrip+on Large system of par.cles with negligible size Boltzmann’s kine.c equa.on Macroscopic descrip+on Con.nuous fluid equa.ons of hydrodynamics (Euler, Navier-­‐Stokes...) N>>1, Nε = α Low density limit α>>1 Fast relaxa.on limit Nε >>1, Nε <<1 d-­‐1