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.

本文利用sigma收敛方法处理线性Boltzmann方程的均质化问题。用涵盖多种物理行为的抽象假设代替经典的碰撞假设系数的周期性假设，我们证明了粒子的密度收敛于漂移扩散方程的解。然后，我们通过计算一些具体的同质化问题（例如周期性的，几乎周期性的一个和其他问题）来说明这种抽象设置。为了实现我们的目标，我们将Krein-Rutman定理用于局部凸空间以及Fredholm替代方案，以解决所谓的校正器问题。