Non-extensive statistical mechanics of a self-gravitating gas

TL;DR

This paper explores the application of Tsallis non-extensive statistics to a self-gravitating gas, deriving analytical expressions for thermodynamic quantities and highlighting ensemble dependence issues.

Contribution

It introduces a generalized statistical framework for self-gravitating systems and provides analytical results for thermodynamics within this non-extensive context.

Findings

Analytical expressions for equation of state and specific heat.

Thermodynamic quantities depend on the non-extensivity parameter in the canonical ensemble.

Ensemble equivalence may not hold in Tsallis statistics for self-gravitating systems.

Abstract

The statistical mechanics of a cloud of particles interacting via their gravitational potentials is an old problem which encounters some issues when the traditional Boltzmann-Gibbs statistics is applied. In this article, we consider the generalized statistics of Tsallis and analyze the statistical and thermodynamical implications for a self-gravitating gas, obtaining analytical and convergent expressions for the equation of state and specific heat in the canonical as well as microcanonical ensembles. Although our results are comparable in both ensembles, it turns out that only in the canonical case the thermodynamic quantities depend explicitly on the non-extensivity parameter, indicating that the question of ensemble equivalence for Tsallis statistics must be further reviewed.