A model for slowing particles in random media

We present a simple model in dimension $ d\geq 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 $ \dot{V} = -\frac{ {\kappa}}{\epsilon} S(|V|) V $, where $ V $ is the velocity of the point particle, $ {\kappa} $ is a positive constant, $ \epsilon $ 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 $ \epsilon $ 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 $ {\delta} $ function in $ v = 0 $; this contribution guarantees the conservation of mass for the limit equation.