1305.3397 (Thierry Bodineau et al.)

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

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond

We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems 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 $N \varepsilon^{d-1} \to \infty$ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium. Our proof relies 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.

View original: http://arxiv.org/abs/1305.3397