Homogenization of the Linear Boltzmann Equation in a Domain with a Periodic Distribution of Holes

Abstract

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing, i.e., that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially quickly as $t\to+\infty$. This is at variance with the collisionless case discussed in [E. Caglioti and F. Golse, Comm. Math. Phys., 236 (2003), pp. 199–221], where the total mass decays as $C/t$ as $t\to+\infty$.