Fractal Structures and Scaling Laws in the Universe: Statistical Mechanics of the Self-Gravitating Gas

Fractal structures are observed in the universe in two very different ways. Firstly, in the gas forming the cold interstellar medium in scales from 10^{-4} pc till 100 pc. Secondly, the galaxy distribution has been observed to be fractal in scales up to hundreds of Mpc. We give here a short review of the statistical mechanical (and field theoretical) approach developed by us. We consider a non-relativistic self-gravitating gas in thermal equilibrium at temperature T inside a volume V. The statistical mechanics of such system has special features and, as is known, the thermodynamical limit does not exist in its customary form. Moreover, the treatments through microcanonical, canonical and grand canonical ensembles yield different results.We present here for the first time the equation of state for the self-gravitating gas in the canonical ensemble. We find that it has the form p = [N T/ V] f(eta), where p is the pressure, N is the number of particles and \eta \equiv {G m^2 N \over V^{1/3} T}. The N \to\infty and V \to\infty limit exists keeping \eta fixed. We compute the function f(\eta) using Monte Carlo simulations and for small eta analytically. We compute the thermodynamic quantities of the system as free energy, entropy, chemical potential, specific heat, compressibility and speed of sound. We reproduce the well-known gravitational phase transition associated to the Jeans' instability. Namely, a gaseous phase for eta < eta_c and a condensed phase for eta > eta_c. Moreover, we derive the precise behaviour of the physical quantities near the transition. In particular, the pressure vanishes as p \sim(eta_c-eta)^B with B \sim 0.2 and eta_c \sim 1.6 and the energy fluctuations diverge as \sim(eta_c-eta)^{B-1}. The speed of sound decreases monotonically and approaches the value sqrt{T/6} at the transition.Comment: Invited paper to the special issue of the `Journal of Chaos, Solitons and Fractals': `Superstrings, M, F, S...theory', M. S El Naschie and C. Castro, Editors. Latex file, 16 pages plus three .ps figure