# Resistive dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation

## TL;DR

This paper derives the equations governing resistive relativistic magnetohydrodynamics from the Boltzmann-Vlasov equation, extending previous non-resistive models to include finite electric conductivity and new transport phenomena.

## Contribution

It introduces a second-order dissipative MHD framework with finite conductivity derived from kinetic theory, incorporating electric field effects and new transport coefficients.

## Key findings

- Derived resistive relativistic MHD equations from Boltzmann-Vlasov equation.
- Established the relation between electrical conductivity and charge diffusion via Wiedemann-Franz law.
- Showed that the Navier-Stokes limit reproduces Ohm's law.

## Abstract

We derive the equations of motion of relativistic, resistive, second-order dissipative magnetohydrodynamics from the Boltzmann-Vlasov equation using the method of moments. We thus extend our previous work [Phys. Rev. D 98, 076009 (2018)], where we only considered the non-resistive limit, to the case of finite electric conductivity. This requires keeping terms proportional to the electric field $E^\mu$ in the equations of motions and leads to new transport coefficients due to the coupling of the electric field to dissipative quantities. We also show that the Navier-Stokes limit of the charge-diffusion current corresponds to Ohm's law, while the coefficients of electrical conductivity and charge diffusion are related by a type of Wiedemann-Franz law.