Time parameters and Lorentz transformations of relativistic stochastic processes

TL;DR

This paper derives transformation rules for time parameters in relativistic stochastic processes and shows how different parameterizations lead to distinct equilibrium distributions, highlighting the impact of relativistic effects.

Contribution

It introduces new transformation rules for time parameters in relativistic Langevin equations and reveals the resulting differences in equilibrium distributions for proper-time and coordinate-time parameterizations.

Findings

Proper-time parameterized process converges to a modified momentum distribution.

Coordinate-time parameterized process approaches the relativistic Juttner-Maxwell distribution.

Transformation rules impact the equilibrium states of relativistic stochastic processes.

Abstract

Rules for the transformation of time parameters in relativistic Langevin equations are derived and discussed. In particular, it is shown that, if a coordinate-time parameterized process approaches the relativistic Juttner-Maxwell distribution, the associated proper-time parameterized process converges to a modified momentum distribution, differing by a factor proportional to the inverse energy.