The Statistical Mechanics of the Self-Gravitating Gas: Equation of State and Fractal Dimension

TL;DR

This paper thoroughly analyzes the thermodynamic properties of a self-gravitating gas at equilibrium using simulations and mean field theory, revealing a fractal particle distribution with a density-dependent Hausdorff dimension.

Contribution

It provides a comprehensive analysis combining Monte Carlo, mean field, and low density expansions to characterize the equation of state and fractal structure of the self-gravitating gas.

Findings

Equation of state derived without assumptions

Agreement between Monte Carlo and mean field results

Fractal distribution with Hausdorff dimension decreasing with density

Abstract

We provide a complete picture of the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations (MC), analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit, both in the canonical (CE) and in the microcanonical ensemble (MCE) when N, V \to \infty, keeping N/ V^{1/3} fixed. We {\bf compute} the equation of state (we do not assume it as is customary), the entropy, the free energy, the chemical potential, the specific heats, the compressibilities, the speed of sound and analyze their properties, signs and singularities. The MF equation of state obeys a {\bf first order} non-linear differential equation of Abel type. The MF gives an accurate picture in agreement with the MC simulations both in the CE and MCE. The inhomogeneous particle distribution in the ground state suggest a fractal distribution with Haussdorf dimension D with D slowly decreasing with increasing density, 1 \lesssim D < 3.