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

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 $N{\epsilon }^2=1$, $\epsilon \to 0$ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.