BOLTZMANN-GRAD LIMIT OF A HARD SPHERE SYSTEM IN A BOX WITH ISOTROPIC BOUNDARY CONDITIONS

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