Solvable model of a self-gravitating system

TL;DR

This paper presents a solvable effective model for self-gravitating systems, derived from the self-gravitating ring model, enabling analytical thermodynamic analysis and comparison with previous models.

Contribution

The paper introduces a new effective model for self-gravitating systems that can be solved analytically in both ensembles, and compares it with the self-gravitating ring model and Thirring's model.

Findings

The effective model matches well with the SGR model despite approximations.

It provides analytical solutions for equilibrium thermodynamics.

The model relates closely to Thirring's minimal model of self-gravitating systems.

Abstract

We introduce and discuss an effective model of a self-gravitating system whose equilibrium thermodynamics can be solved in both the microcanonical and the canonical ensemble, up to a maximization with respect to a single variable. Such a model can be derived from a model of self-gravitating particles confined on a ring, referred to as the self-gravitating ring (SGR) model, allowing a quantitative comparison between the thermodynamics of the two models. Despite the rather crude approximations involved in its derivation, the effective model compares quite well with the SGR model. Moreover, we discuss the relation between the effective model presented here and another model introduced by Thirring forty years ago. The two models are very similar and can be considered as examples of a class of minimal models of self-gravitating systems.