Statistical Mechanics of the Self-Gravitating Gas: Thermodynamic Limit, Phase Diagrams And Fractal Structures

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 and speed of sound; we analyze their properties, signs and singularities. All physical quantities turn out to depend on a single variable \( \eta \equiv \frac{{Gm^2 N}} {{V^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} T}} \) that is kept fixed in the N → ∞ and V → ∞ limit. The system is in a gaseous phase for η ηT The values of ηT and ηC change by a few percent with the geometry for large N: for spherical symmetry and N = ∞ (MF), we find ηC = 1.561764…, while the Monte Carlo simulations for cubic geometry yields ηC ≃ 1.540. In mean field and spherical symmetry cy diverges as (ηC •η )•1/2 for η ↑ ηC while cp and KT diverge as (ηC •η) •1 for η ↑ η o = 1.51024 …. The 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 ηMC = 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. The MF equation of state in a sphere, f(η ), obeys a first order non-linear differential equation of first kind Abel’s type. The MF gives an extremely accurate picture in agreement with the MC simulations both in the CE and MCE. Since we perform the MC simulations on a cubic geometry they describe an isothermal cube while the MF calculations describe an isothermal sphere.