Abstract. The paper deals with the diffusion limit of the initial-boundary value problem for the multi-dimensional semiconductor Boltzmann-Poisson system. Here, we generalize the one dimensional results obtained in [6] to the case of several dimensions using global renormalized solutions. The method of moments and a velocity averaging lemma are used to prove the convergence of the renormalized solutions to the semiconductor Boltzmann-Poisson system towards a global weak solution of the Drift-Diffusion-Poisson model.

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This paper deals with the diffusion approximation of the Boltzmann equation for semiconductors in the presence of spatially oscillating electrostatic potential. When the oscillation period is of the same order of magnitude as the mean free path, the asymptotics leads to the driftdiffusion equation with a homogenized electrostatic potential and a diffusion matrix involving the small-scale information. The convergence is proven rigorously for Boltzmann statistics, while it is incomplete for Fermi-Dirac statistics. Introduction.The drift-diffusion equation is a standard model of particle transport in many applications such as neutron transport [17,18], plasmas [14,22], semiconductors [25,26,33,34,41,42,40,44], gas discharges [39], etc. The driftdiffusion model, very well suited for numerical simulations, is designed to describe the macroscopic behavior of the device. It consists of a mass balance equation relying on the particle and current densities,

This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.