Homogenization of linear Boltzmann equations in the context of algebras with mean value

Abstract

The paper deals with the homogenization of linear Boltzmann equations by the means of the sigma-convergence method. Replacing the classical periodicity hypothesis on the coefficients of the collision operator by an abstract assumption covering a great variety of physical behaviours, we prove that the density of the particles converges to the solution of a drift-diffusion equation. We then illustrate this abstract setting by working out a few concrete homogenization problems such as the periodic one, the almost periodic one and others. To achieve our goal, we use the Krein–Rutman theorem for locally convex spaces together with the Fredholm alternative to solve the so-called corrector problem.