The Brownian motion as the limit of a deterministic system of hard-spheres

Abstract

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 \(\varepsilon \) simultaneously goes to \(0\), in the fast relaxation limit \(\alpha = N\varepsilon ^{d-1}\rightarrow \infty \) (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.