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.

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The diffusion approximation for a Boltzmann–Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful [Formula: see text]-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.

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.