Statistical Mechanics of the Self-Gravitating Gas: I. Thermodynamic Limit and Phase Diagram

Abstract

We provide a complete picture to the selfgravitating 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) --> infinity, 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, compressibi- lities 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--> infinity and V --> infinity limit. The system is in a gaseous phase for eta < eta_T and collapses into a dense objet for eta > η_T in the CE with the pressure becoming large and negative. At eta simeq eta_T the isothermal compressibility diverges. Our Monte Carlo simulations yield eta_T simeq 1.515. PV/[NT] = f(eta) and all physical magni- tudes exhibit a square root branch point at eta = eta_C > eta_T. The MF for spherical symmetry yields eta_C = 1.561764.. while Monte Carlo on a cube yields eta_C simeq 1.540.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 gas phase.Both collapse phase transitions (CE and MCE) are of zeroth order since the Gibbs free energy jumps at the transitions. f(eta), obeys in MF a first order non-linear differential equation of first kind Abel's type.The MF gives an extremely accurate picture in agreement with Monte Carlo both in the CE and MCE.