Résumé en anglais

In this Note we present the main results from the recent work arxiv:1107.3251, which answers several conjectures raised fifty years ago by Kac. There Kac introduced a many-particle stochastic process (now denoted as Kac's master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in \cite{kac}: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the $H$-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.