From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

Abstract

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of $N$ hard spheres of diameter $\varepsilon$ in two space dimensions, when~$N\to \infty$, $\varepsilon \to 0$, $N\varepsilon =\alpha \to \infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy \cite{lanford}, and on the pruning procedure developed in \cite{BGSR1} to improve the convergence time. The main novelty here is that uniform $L^2$ a priori estimates combined with a subtle symmetry argument provide a useful cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.