Diffusion limit of kinetic equations for multiple species charged particles

TL;DR

This paper rigorously derives the Poisson-Nernst-Planck system as a macroscopic limit of a kinetic VPFP model for multi-species charged particles, providing mathematical justification for using PNP in ionic solutions.

Contribution

It proves the convergence of solutions from a kinetic VPFP system to the macroscopic PNP system for multiple ionic species.

Findings

Global solutions of VPFP converge to PNP solutions as parameters tend to zero.

Mathematical justification for PNP as a limit of kinetic models.

Applicable to dilute ionic solutions with multiple species.

Abstract

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.