Relativistic generalization of Brownian Motion

TL;DR

This paper develops a relativistic extension of Brownian motion, ensuring Lorentz invariance of the equilibrium distribution, which leads to a non-covariant noise term due to the entanglement with the force.

Contribution

It introduces a relativistic framework for Brownian motion that maintains Lorentz invariance of the equilibrium distribution, revealing the non-covariant nature of the noise.

Findings

The noise term's transformation depends on Lorentz invariance requirements.

Lorentz invariance of the equilibrium distribution constrains the noise and force relationship.

The noise in relativistic Brownian motion is not a covariant quantity.

Abstract

The relativistic generalization of the Brownian motion is discussed. We show that the transformation property of the noise term is determined by requiring for the equilibrium distribution function to be Lorentz invariant, such as the J\"uttner distribution function. It is shown that this requirement generates an entanglement between the force term and the noise so that the noise itself should not be a covariant quantity.