Theory of Relativistic Brownian Motion: The (1+3)-Dimensional Case

TL;DR

This paper develops a covariant relativistic Brownian motion theory in (1+3) dimensions, analyzing stochastic dynamics, discretization rules, and stationary distributions, with numerical results illustrating relativistic effects on particle displacement.

Contribution

It introduces a relativistically covariant Langevin framework in (1+3) dimensions, clarifies the discretization rule dilemma, and demonstrates the unique stationary Maxwell-Boltzmann distribution in this context.

Findings

Post-point discretization rule yields correct relativistic Maxwell-Boltzmann statistics.

Relativistic velocity effects are more pronounced in three spatial dimensions.

Numerical results show asymptotic mean square displacement of relativistic particles.

Abstract

A theory for (1+3)-dimensional relativistic Brownian motion under the influence of external force fields is put forward. Starting out from a set of relativistically covariant, but multiplicative Langevin equations we describe the relativistic stochastic dynamics of a forced Brownian particle. The corresponding Fokker-Planck equations are studied in the laboratory frame coordinates. In particular, the stochastic integration prescription, i.e. the discretization rule dilemma, is elucidated (pre-point discretization rule {\it vs.} mid-point discretization rule {\it vs.} post-point discretization rule). Remarkably, within our relativistic scheme we find that the post-point rule (or the transport form) yields the only Fokker-Planck dynamics from which the relativistic Maxwell-Boltzmann statistics is recovered as the stationary solution. The relativistic velocity effects become distinctly more pronounced by going from one to three spatial dimensions. Moreover, we present numerical results for the asymptotic mean square displacement of a free relativistic Brownian particle moving in (1+3) dimensions.