Equilibrium Distribution of Heavy Quarks in Fokker-Planck Dynamics

TL;DR

This paper generalizes Einstein's relation within Fokker-Planck dynamics for systems with multiple dimensions, characterizes conditions for different equilibrium distributions, and applies findings to charm quark behavior in thermal plasmas.

Contribution

It provides a comprehensive analysis of equilibrium distributions in Fokker-Planck systems, including conditions for Boltzmann, J"uttner, and Tsallis distributions, with practical applications to relativistic plasmas.

Findings

Tsallis distribution accurately describes equilibrium for charm quarks in plasma

Derived conditions for when equilibrium is Boltzmann or Tsallis

Practical method to determine diffusion coefficients for desired distributions

Abstract

We obtain within Fokker-Planck dynamics an explicit generalization of Einstein's relation between drag, diffusion and equilibrium distribution for a spatially homogeneous system, considering both the transverse and longitudinal diffusion for dimension n>1. We then provide a complete characterization of when the equilibrium distribution becomes a Boltzmann/J"uttner distribution, and when it satisfies the more general Tsallis distribution. We apply this analysis to recent calculations of drag and diffusion of a charm quark in a thermal plasma, and show that only a Tsallis distribution describes the equilibrium distribution well. We also provide a practical recipe applicable to highly relativistic plasmas, for determining both diffusion coefficients so that a specific equilibrium distribution will arise for a given drag coefficient.