One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model

TL;DR

This paper develops a microscopic collision model for one-dimensional Brownian motion, analyzing both nonrelativistic and relativistic cases, and identifies stationary momentum distributions through integral criteria derived from collision kinematics.

Contribution

It introduces a self-consistent integral criterion approach to determine stationary distributions for microscopic collision models in both nonrelativistic and relativistic regimes.

Findings

Maxwellian distribution confirmed in nonrelativistic case

Stationary relativistic distribution differs from Jüttner distribution

Integral criteria based on collision kinematics are effective

Abstract

We study a simple microscopic model for the one-dimensional stochastic motion of a (non)relativistic Brownian particle, embedded into a heat bath consisting of (non)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the nonrelativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard J\"uttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.