Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma)

Abstract

Consider a microscopic system of $N$ hard spheres that are initially independent (modulo the exclusion condition on particle positions) and identically distributed in ${\mathbb{R}}^3$. When the number $N$ of particles goes to infinity and the diameter $\epsilon$ of the particles goes to zero, and under the weak density assumption $N{\epsilon }^2=1$, it has been known since the work of Lanford that the empirical measure for the particles converges to the solution of the Boltzmann equation in a short time interval. In particular, the particles remain dynamically independent, in this limit and in the short time interval where the correlations induced by their collisions remain under control. In a recent work, Y. Deng, Z. Hani and X. Ma successfully obtained the same convergence result in arbitrary large time; more precisely, the convergence result holds for any time interval on which the Boltzmann equation has a regular solution. In this note, we explain a few elements of their proof.

Bourbaki Seminar, January 2026