Derivation of Ohm's law from the kinetic equations

TL;DR

This paper provides a formal derivation of Ohm's law and related models in magnetohydrodynamics starting from kinetic equations, using physical scalings and moment methods as the Knudsen number approaches zero.

Contribution

It introduces a rigorous derivation of MHD models, including Ohm's law and the Hall effect, from the Vlasov-Maxwell-Boltzmann system, connecting kinetic theory to macroscopic equations.

Findings

Derivation of Ohm's law from kinetic equations.

Derivation of the Hall effect and other limit models.

Includes both compressible and incompressible MHD models.

Abstract

The goal of this article is to give a formal derivation of Ohm's law of Magnetohydrodynamics (MHD) starting from the Vlasov-Maxwell-Boltzmann system. The derivation is based on various physical scalings and the moment methods when the Knudsen number goes to zero. We also give a derivation of the so-called Hall effect as well as other limit models such as the Navier-Stokes-Maxwell system. Our results include both the compressible and incompressible MHD models.