Multidimensional high-field limit of the electrostatic Vlasov–Poisson–Fokker–Planck system

In this paper, the high-field limit of the Vlasov–Poisson–Fokker–Planck system for charged particles is rigorously derived. The first result is obtained in any space dimension by using modulated energy techniques. It requires the smoothness of the solutions of the limit problem. In dimension 2, it is possible to handle more general data by using methods developed for a diagonal defect measures theory. The convergence of the concentration of particles is obtained in the space of bounded measures. In both cases, the limit of the sequence of densities of distribution functions is shown to solve a nonlinear system of partial differential equations which is related to Ohmʹs law