FRACTAL STRUCTURES AND SCALING LAWS IN THE UNIVERSE: STATISTICAL MECHANICS OF THE SELF-GRAVITATING GAS

Abstract

Fractal structures are observed in the universe in two very different ways. Firstly, in the gas forming the cold interstellar medium in scales from 10-4pc till 100pc. 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 for the cold interstellar medium (ISM) and large structure of the universe. 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 = [NT/V] f (rl), where p is the pressure , N is the number of particles and n = -VT/ . The N -+ 00 and V -4 oo limit exists keeping rl fixed. We compute the function f (n) using Monte Carlo simulations and for small n, 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 q rlc. Moreover , we derive the precise behaviour of the physical quantities near the transition . In particular , the pressure vanishes asp (p, 77)o with B 0.2 and n, 1.6 and the energy fluctuations diverge as (% -u )B-1. The speed of sound decreases monotonically with n and approaches the value VI T6 at the transition.