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

TL;DR

This paper discusses the derivation of the Boltzmann equation from the microscopic dynamics of hard spheres, extending previous short-time results to arbitrary large times under certain regularity conditions.

Contribution

It explains key elements of Deng, Hani, and Ma's proof that extends the convergence of particle systems to the Boltzmann equation for large times.

Findings

Convergence of particle system to Boltzmann equation for large times

Extension of Lanford's short-time result to arbitrary large times

Conditions under which the derivation holds

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 $\varepsilon$ of the particles goes to zero, and under the weak density assumption $N\varepsilon^2=1$, 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.