Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation

TL;DR

This paper derives the relativistic Langevin equation from a microscopic collision model, revealing that the relativistic stochastic force remains white noise but is no longer Gaussian, with explicit friction and diffusion coefficients provided.

Contribution

It extends the microscopic derivation of the Langevin equation to the relativistic regime, showing the non-Gaussian nature of the stochastic force.

Findings

Relativistic stochastic force is white noise but non-Gaussian.

Explicit formulas for friction and diffusion coefficients.

Non-relativistic limit recovers known Langevin equation.

Abstract

The Langevin equation (LE) for the one-dimensional relativistic Brownian motion is derived from a microscopic collision model. The model assumes that a heavy point-like Brownian particle interacts with the lighter heat bath particles via elastic hard-core collisions. First, the commonly known, non-relativistic LE is deduced from this model, by taking into account the non-relativistic conservation laws for momentum and kinetic energy. Subsequently, this procedure is generalized to the relativistic case. There, it is found that the relativistic stochastic force is still $\gd$-correlated (white noise) but does \emph{no} longer correspond to a Gaussian white noise process. Explicit results for the friction and momentum-space diffusion coefficients are presented and discussed.