Boltzmann-Grad limit of a hard sphere system in a box with diffusive boundary conditions

TL;DR

This paper rigorously derives the Boltzmann equation for a hard sphere system in a bounded domain with diffusive boundary conditions, demonstrating convergence of particle dynamics to kinetic theory in a specific scaling limit.

Contribution

It extends Lanford's theorem to systems with diffusive boundary conditions in a bounded domain, providing a rigorous derivation of the Boltzmann equation in this setting.

Findings

Convergence of the particle system to the Boltzmann equation in the Boltzmann-Grad limit.

Extension of Lanford's theorem to diffusive boundary conditions.

Validation of kinetic theory in bounded domains with stochastic boundary interactions.

Abstract

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with diffuse reflection boundary conditions. We consider a system of $N$ hard spheres of diameter $\epsilon$ in a box $\Lambda := [0, 1] \times (R/Z)^2$. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, and conserving kinetic energy. We prove that the first marginal of the process converges in the scaling $N\epsilon^2 = 1$, $\epsilon\rightarrow 0$ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.