Thermodynamics of the self-gravitating ring model

TL;DR

This paper analyzes the thermodynamics and phase transitions of the self-gravitating ring model, revealing ensemble inequivalence, negative specific heat regions, and a tricritical point, with a new iterative method for stable equilibrium computation.

Contribution

It provides the first detailed phase diagram of the SGR model, highlighting ensemble differences and introducing a novel iterative method for equilibrium mass distribution.

Findings

Existence of a negative specific heat phase in microcanonical ensemble.

Phase transition changes from second to first order at a tricritical point.

A new iterative method ensures entropy increase and convergence to equilibrium.

Abstract

We present the phase diagram, in both the microcanonical and the canonical ensemble, of the Self-Gravitating-Ring (SGR) model, which describes the motion of equal point masses constrained on a ring and subject to 3D gravitational attraction. If the interaction is regularized at short distances by the introduction of a softening parameter, a global entropy maximum always exists, and thermodynamics is well defined in the mean-field limit. However, ensembles are not equivalent and a phase of negative specific heat in the microcanonical ensemble appears in a wide intermediate energy region, if the softening parameter is small enough. The phase transition changes from second to first order at a tricritical point, whose location is not the same in the two ensembles. All these features make of the SGR model the best prototype of a self-gravitating system in one dimension. In order to obtain the stable stationary mass distribution, we apply a new iterative method, inspired by a previous one used in 2D turbulence, which ensures entropy increase and, hence, convergence towards an equilibrium state.