Relativistic diffusion

TL;DR

This paper develops a relativistic diffusion model using stochastic differential equations, explores their solutions in electromagnetic fields, and investigates equilibrium distributions, including connections to quantum mechanics and heavy ion collisions.

Contribution

It introduces explicit Langevin equations for relativistic diffusion, analyzes equilibrium distributions, and links the process to quantum mechanics on a hyperboloid.

Findings

Explicit form of momentum distribution derived.

Tsallis distribution satisfies diffusion approximation restrictions.

Relativistic diffusion relates to quantum mechanics on hyperboloids.

Abstract

We discuss a relativistic diffusion in the proper time in an approach of Schay and Dudley. We derive (Langevin) stochastic differential equations in various coordinates.We show that in some coordinates the stochastic differential equations become linear. We obtain momentum probability distribution in an explicit form.We discuss a relativistic particle diffusing in an external electromagnetic field. We solve the Langevin equations in the case of parallel electric and magnetic fields. We derive a kinetic equation for the evolution of the probability distribution.We discuss drag terms leading to an equilibrium distribution.The relativistic analog of the Ornstein-Uhlenbeck process is not unique. We show that if the drag comes from a diffusion approximation to the master equation then its form is strongly restricted. The drag leading to the Tsallis equilibrium distribution satisfies this restriction whereas the one of the Juettner distribution does not. We show that any function of the relativistic energy can be the equilibrium distribution for a particle in a static electric field. A preliminary study of the time evolution with friction is presented. It is shown that the problem is equivalent to quantum mechanics of a particle moving on a hyperboloid with a potential determined by the drag. A relation to diffusions appearing in heavy ion collisions is briefly discussed.