Relativistic diffusion with friction on a pseudoriemannian manifold

TL;DR

This paper investigates relativistic diffusion processes on a pseudoriemannian manifold, analyzing stochastic equations, equilibrium states, and specific spacetime examples, revealing exponential growth and bounded behaviors of physical quantities.

Contribution

It introduces a detailed analysis of relativistic diffusion with friction on pseudoriemannian manifolds, connecting stochastic equations to physical equilibria and specific spacetime models.

Findings

Energy and angular momentum grow exponentially without equilibrium.

Diffusion processes can reach equilibrium with bounded energy and angular momentum.

Relativistic diffusion in de Sitter space is equivalent to Minkowski diffusion with temperature proportional to the de Sitter radius.

Abstract

We study a relativistic diffusion equation on the Riemannian phase space defined by Franchi and Le Jan. We discuss stochastic Ito (Langevin) differential equations (defining the diffusion) as a perturbation by noise of the geodesic equation. We show that the expectation value of the angular momentum and the energy grow exponentially fast. We discuss drifts leading to an equilibrium. It is shown that the diffusion process corresponding to the Juettner or quantum equilibrium distributions has a bounded expectation value of angular momentum and energy. The energy and the angular momentum tend exponentially fast to their equilibrium values. As examples we discuss a particle in a plane fronted gravitational wave and a particle in de Sitter universe. It is shown that the relativistic diffusion of momentum in de Sitter space is the same as the relativistic diffusion on the Minkowski mass-shell with the temperature proportional to the de Sitter radius.