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Diffusion limit of kinetic equations for multiple species charged particles. (English) Zbl 1308.35311
Summary: In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson-Nernst-Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov-Poisson-Fokker-Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.

MSC:
35Q84 Fokker-Planck equations
35Q83 Vlasov equations
82C70 Transport processes in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35D30 Weak solutions to PDEs
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