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

Corentin Le Bihan (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 Nε^2 = 1, ε→0 to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

Mots clés: Low density; Boltzmann-Grad limit; Hard sphere; Stochastic boundary condition

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Probabilités [math.PR]

Corentin Le Bihan. Boltzmann-Grad limit of a hard sphere system in a box with diffusive boundary conditions. 2021. ⟨hal-03194097⟩