Statistical Mechanics of the Self-Gravitating Gas. Thermodynamic Limit, Phase Diagrams Local Physical Magnitudes and Fractal Structures

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) → ∞, keeping N/V ^1/3 fixed. We compute the equation of state (we do not assume it as is customary), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibilities, speed of sound and particle density; we analyze their properties, signs and singularities. All physical quantities turn out to depend on a single variable $\eta \equiv \frac{{Gm^2 N}} {{V^{1/3} T}}$ that is kept fixed in the N → ∞ and V → ∞ limit . Th e system is in a gaseous phase for η < ητ and collapses into a dense objet for η > ητ in the CE with the pressure becoming large and negative. At η ≃ ητ the isothermal compressibility diverges. This gravitational phase transition is associated to the Jeans’ instability. Our Monte Carlo simulations yield ητ ≃: 1.515. PV/[NT] = f(η) and all physical magnitudes exhibit a square root branch point at η = η c > ητ. The values of ητ and η c change by a few percent with the geometry for large N: for spherical symmetry and N = ∞ (MF), we find η c =1.561764 ... while th e Monte Carlo simulations for cubic geometry yields η c ≃ 1.540. Th e function f(η) 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 (η)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.