Homogenization and diffusion approximation of the Vlasov–Poisson–Fokker–Planck system: A relative entropy approach

Abstract

We are concerned with the analysis of the approximation by diffusion and homogenization of a Vlasov–Poisson–Fokker–Planck system. Here we generalize the convergence result of (Comm. Math. Sci. 8 (2010), 463–479) where the same problem is treated without the oscillating electrostatic potential and we extend the one dimensional result of (Ann. Henri Poincaré 17 (2016), 2529–2553) to the case of several space dimensions. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Vlasov–Poisson–Fokker–Planck system to a homogenized Drift-Diffusion-Poisson system.