Limite de diffusion lin\'eaire pour un syst\`eme d\'eterministe de sph\`eres dures

We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\epsilon$ simultaneously goes to $0,$ in the fast relaxation limit $N{\epsilon }^{d-1}\to \mathrm{\infty }$ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use Boltzmann’s kinetic theory as an intermediate level of description for the gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.