Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation

TL;DR

This paper investigates the relativistic Fokker-Planck equation, demonstrating its convergence to the classical form as the speed of light increases and establishing exponential convergence to equilibrium under certain conditions.

Contribution

It proves the Newtonian limit of the relativistic Fokker-Planck equation and establishes exponential trend to equilibrium with rate estimates depending on the speed of light.

Findings

Solutions converge to non-relativistic solutions as c→∞

Exponential trend to equilibrium for spatially homogeneous solutions at low temperature

Uniform convergence rate estimates depending on the speed of light

Abstract

The relativistic Fokker-Planck equation, in which the speed of light $c$ appears as a parameter, is considered. It is shown that in the limit $c\to\infty$ its solutions converge in $L^1$ to solutions of the non-relativistic Fokker-Planck equation, uniformly in compact intervals of time. Moreover in the case of spatially homogeneous solutions, and provided the temperature of the thermal bath is sufficiently small, exponential trend to equilibrium in $L^1$ is established. The dependence of the rate of convergence on the speed of light is estimated. Finally, it is proved that exponential convergence to equilibrium for all temperatures holds in a weighted $L^2$ norm.