Non-linear relativistic diffusions

TL;DR

This paper develops a non-linear relativistic diffusion model incorporating quantum statistics, analyzing invariant and covariant equations, stationary solutions, and thermodynamic properties of diffusing particles.

Contribution

It introduces a non-linear generalization of relativistic diffusion with quantum statistical effects, exploring invariant and covariant forms and their thermodynamic implications.

Findings

Existence of stationary solutions beyond equilibrium distributions.

Relative entropy decreases monotonically over time.

Relativistic invariance constrains admissible non-linear diffusion equations.

Abstract

We obtain a non-linear generalization of the relativistic diffusion of particles with spin. We discuss diffusion equations whose non-linearity is a consequence of quantum statistics. We show that the assumptions of the relativistic invariance and an interpretation of the solution as a probability distribution substantially restrict the class of admissible non-linear diffusion equations. We consider relativistic invariant as well as covariant frame dependent diffusion equations with a drift. In the latter case we show that there can exist stationary solutions of the diffusion equation besides the equilibrium solution corresponding to the quantum or Tsallis distributions. We define the relative entropy as a function of the diffusion probability and prove that it is monotonically decreasing in time when the diffusion tends to the equilibrium. We discuss its relation to the thermodynamic behavior of diffusing particles.