Fokker-Planck type equations for a simple gas and for a semi-relativistic Brownian motion from a relativistic kinetic theory

TL;DR

This paper derives covariant Fokker-Planck equations from relativistic kinetic theory for a simple gas and semi-relativistic Brownian motion, analyzing friction and diffusion in relativistic contexts.

Contribution

It introduces a covariant Fokker-Planck framework for relativistic gases and Brownian motion, including explicit expressions for friction and diffusion tensors.

Findings

Derived relativistic Fokker-Planck equations for gases and Brownian motion.

Obtained expressions for relativistic friction and diffusion tensors.

Analyzed non-relativistic and ultra-relativistic limits for constant cross-section cases.

Abstract

A covariant Fokker-Planck type equation for a simple gas and an equation for the Brownian motion are derived from a relativistic kinetic theory based on the Boltzmann equation. For the simple gas the dynamic friction four-vector and the diffusion tensor are identified and written in terms of integrals which take into account the collision processes. In the case of Brownian motion, the Brownian particles are considered as non-relativistic whereas the background gas behaves as a relativistic gas. A general expression for the semi-relativistic viscous friction coefficient is obtained and the particular case of constant differential cross-section is analyzed for which the non-relativistic and ultra relativistic limiting cases are calculated.