Asymptotic analysis for the Generalized Relativistic Langevin Equation

TL;DR

This paper analyzes a non-Markovian relativistic Langevin equation, showing its equivalence to a Markovian system with exponential memory kernels, and establishes its ergodic properties and limits to classical dynamics.

Contribution

It introduces an analysis of the generalized relativistic Langevin equation with exponential kernels, proving well-posedness, ergodicity, and connections to classical and relativistic limits.

Findings

Proves polynomial ergodicity and algebraic convergence rate.

Shows equivalence to a Markovian system with auxiliary variables.

Recovers classical and relativistic Langevin dynamics in respective limits.

Abstract

In this paper, we study a non-Markovian generalized relativistic Langevin equation (GRLE). We show that when the memory kernel is a sum of exponentials, the GRLE is equivalent to a Markovian system with added variables. We establish the well-posedness and polynomial ergodicity, obtaining an algebraic rate of convergence to the unique Gibbs distribution. From the Markovian GRLE, we recover the relativistic underdamped Langevin dynamics in a small-noise limit, as well as the classical (non-relativistic) generalized Langevin dynamics in the Newtonian limit.