Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials

Abstract

We study a system of interacting particles in the presence of the relativistic kinetic energy, external confining potentials, singular repulsive forces as well as a random perturbation through an additive white noise. In comparison with the classical Langevin equations that are known to be exponentially attractive toward the unique statistically steady states, we find that the relativistic systems satisfy algebraic mixing rates of any order. This relies on the construction of Lyapunov functions adapting to previous literature developed for irregular potentials. We then explore the Newtonian limit as the speed of light tends to infinity and establish the validity of the approximation of the solutions by the Langevin equations on any finite time window.