  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 $\varepsilon$ 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 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.
