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

2021/04/09 by Bihan, Corentin Le
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2104.04354

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 ε in a box Λ:= [0, 1] × (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 Nε² = 1, ε→ 0 to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

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