Diffusion in Curved Spacetimes

TL;DR

This paper derives the relativistic diffusion equation in curved spacetimes, providing a microscopic basis for Eckart's heat flux model and analyzing Brownian motion behavior in static gravitational fields.

Contribution

It introduces a derivation of the Fokker-Planck equation for diffusion in curved spacetimes, linking microscopic processes to relativistic heat transfer models.

Findings

Derivation of the relativistic diffusion equation in curved spacetimes.

Connection of the equation to Eckart's relativistic heat flux.

Asymptotic analysis of Brownian motion in static spacetimes.

Abstract

Using simple kinematical arguments, we derive the Fokker-Planck equation for diffusion processes in curved spacetimes. In the case of Brownian motion, it coincides with Eckart's relativistic heat equation (albeit in a simpler form), and therefore provides a microscopic justification for his phenomenological heat-flux ansatz. Furthermore, we obtain the small-time asymptotic expansion of the mean square displacement of Brownian motion in static spacetimes. Beyond general relativity itself, this result has potential applications in analogue gravitational systems.