A model for slowing particles in random media

Abstract

We present a simple model in dimension d ≥ 2 for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle, a particle is slowed down according to the law V˙ = − κ ǫS(|V |)V , where V is the velocity of the point particle, κ is a positive constant, ǫ is the radius of the obstacle and S(|V |) is a given slowing profile. With this choice, the slowing rate in the obstacles is such that the variation of speed at each crossing is of order 1. We study the asymptotic limit of the particle system when ǫ vanishes and the mean free path of the point particles stays finite. We prove the convergence of the point particles density measure to the solution of a kinetic-like equation with a collision term which includes a contribution proportional to a δ function in v = 0; this contribution guarantees the conservation of mass for the limit equation.