Diffusion and Homogenization Approximation for Semiconductor Boltzmann–poisson System

Abstract

Abstract. We are concerned with the study of the diffusion and homogenization approximation of the Boltzmann-Poisson system in presence of a spatially oscillating electrostatic potential. By analyzing the relative entropy, we prove uniform energy estimate for well prepared boundary data. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Boltzmann equation (coupled to Poisson) to a homogenized Drift-Diffusion-Poisson system.

Homogenization and Hydrodynamic Limit for Fermi‐Dirac Statistics Coupled to a Poisson Equation

“…As in [19], the proof uses the averaging lemma and control from the entropy dissipation. Due to the presence of oscillations and the low integrability bound on f " , we have to use the two-scale convergence and two-scale Young measures [7,8,26].…”

“…0. Classically, this is done by using a duality method or by using a double-scale weak formulation [1,19]. However, this does not work here.…”

“…Here we extend the analysis done in [19] to the case of a nonlinear collision operator. In addition to the averaging lemma and control from entropy dissipation used in [19], here we use two-scale Young measures and renormalization techniques to prove the convergence. This result rigorously justifies the formal analysis of [3].…”

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

“…To do this, we identify each bounded sequence η ε := η ε (t, x, v) of L 2 loc (Ω T ) to its twoscale Riesz's representation [22]. We denote this two-scale function byη ε := η ε (t, x, y, v) ∈ L 2 loc (Ω T ; L 2 # (Y )) in the sense that for all ψ ∈ D(Ω T ; C # (Y )),…”

“…Progressively, different fluid models have been obtained as a hydrodynamic limit of kinetic equations for prescribed macroscopic potential [25,26]. Nowadays, more attention is paid to the variations of the potential by coupling the distribution with the Poisson equation [4,5,11,15,16,18] and recently by adding a potential with microscopic variations in the same order of the mean free path of the diffusive operator, leading to phenomena of homogenization [10,17,19,22,30]. The concept of relative entropy dissipation is useful to approximate the solution of the kinetic model to its corresponding equilibrium state [6,7,12,21,24,28,29].…”