Relativistic stochastic mechanics II: Reduced Fokker-Planck equation in curved spacetime

TL;DR

This paper derives a covariant reduced Fokker-Planck equation in curved spacetime, clarifies distribution functions, and connects relativistic kinetic theory with equilibrium states and Einstein relations.

Contribution

It presents a unified derivation of the covariant Fokker-Planck equation and clarifies the relationship between different distribution functions in relativistic stochastic mechanics.

Findings

Derived the covariant Fokker-Planck equation in curved spacetime.

Clarified the relationship between various distribution functions.

Established a covariant Einstein relation in relativistic context.

Abstract

The general covariant Fokker-Planck equations associated with the two different versions of covariant Langevin equation in Part I of this series of work are derived, both lead to the same reduced Fokker-Planck equation for the non-normalized one particle distribution function (1PDF). The relationship between various distribution functions is clarified in this process. Several macroscopic quantities are introduced by use of the 1PDF, and the results indicate an intimate connection with the description in relativistic kinetic theory. The concept of relativistic equilibrium state of the heat reservoir is also clarified, and, under the working assumption that the Brownian particle should approach the same equilibrium distribution as the heat reservoir in the long time limit, a general covariant version of Einstein relation arises.