Statistical Mechanics of the self-gravitating gas: thermodynamic limit, phase diagrams and fractal structures

TL;DR

This paper analyzes the thermodynamic behavior of a self-gravitating gas at equilibrium using simulations and analytic methods, revealing phase transitions, singularities, and fractal structures dependent on a key parameter.

Contribution

It provides a comprehensive analysis of the self-gravitating gas's thermodynamics, including phase diagrams and the role of the variable eta, using multiple methods.

Findings

Identifies a critical eta_T where the gas collapses and compressibility diverges.

Discovers a second Riemann sheet in the equation of state relevant in the microcanonical ensemble.

Shows collapse phase transitions are of zeroth order with jumps in free energy.

Abstract

We provide a complete picture to the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit in the grand canonical (GCE), canonical (CE) and microcanonical (MCE) ensembles when (N, V) -> infty, keeping N/V^{1/3} fixed. We compute the equation of state (we do not assume it as is customary in hydrodynamics), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibilities and speed of sound; we analyze their properties, signs and singularities. All physical quantities turn out to depend on a single variable eta = G m^2 N /[V^{1/3} T} that is kept fixed in the N -> infty and V -> infty limit. The system is in a gaseous phase for eta < eta_T and collapses into a dense object for eta > eta_T in the CE with the pressure becoming large and negative. At eta \simeq eta_T the isothermal compressibility diverges and the gas collapses. Our Monte Carlo simulations yield eta_T \simeq 1.515. We find that PV/[NT] = f(eta). The function f(eta) has a second Riemann sheet which is only physically realized in the MCE. In the MCE, the collapse phase transition takes place in this second sheet near eta_{MC} = 1.26 and the pressure and temperature are larger in the collapsed phase than in the gaseous phase. Both collapse phase transitions (in the CE and in the MCE) are of zeroth order since the Gibbs free energy has a jump at the transitions.