Diffusion in the special theory of relativity

TL;DR

This paper extends Markovian diffusion theory to special relativity by developing a Lorentz-invariant diffusion equation and deriving solutions, including the Jüttner distribution, for relativistic particles in external fields.

Contribution

It introduces a Lorentz-invariant diffusion framework on Lorentzian manifolds and derives a generalized relativistic Kramers equation with analytical solutions.

Findings

The diffusion equation is Lorentz invariant.

Stationary solution is the Jüttner distribution.

Analytical solution for force-free relativistic diffusion.

Abstract

The Markovian diffusion theory is generalized within the framework of the special theory of relativity using a modification of the mathematical calculus of diffusion on Riemannian manifolds (with definite metric) to describe diffusion on Lorentzian manifolds with an indefinite metric. A generalized Langevin equation in the fiber space of position, velocity and orthonormal velocity frames is defined from which the generalized relativistic Kramers equation in the phase space in external force fields is derived. The obtained diffusion equation is invariant under Lorentz transformations and its stationary solution is given by the J\"{u}ttner distribution. Besides a non-stationary analytical solution is derived for the example of force-free relativistic diffusion.