Fractal structures and scaling laws in the universe. Statistical mechanicsof 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 100pc. Secondly, the galaxy distribution has been observed to b e fractal in scales up to hundreds of Mpc. We give here a short review of t he 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 syst em has special features and, as is known, the thermodynamical limit does no t exist in its customary form. Moreover, the treatments through microcanoni cal, canonical and grand canonical ensembles yield different results. We pr esent here for the first time the equation of state for the self-gravitatin g gas in the canonical ensemble. We find that it has the form p = [NT/V]f(e ta), where p is the pressure, N is the number of particles and eta = Gm(2)N /(VT)-T-1/3. The N --> infinity and V --> infinity limit exists keeping eta fixed. We compute the function f(eta) using Monte Carlo simulations and fo r small eta, analytically. We compute the thermodynamic quantities of the s ystem as free energy, entropy, chemical potential, specific heat, compressi bility and speed of sound. We reproduce the well-known gravitational phase transition associated to the Jeans' instability. Namely, a gaseous phase fo r 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 pa rticular, the pressure vanishes as p similar to (eta(c) - eta)(B) with B si milar to 0.2 and eta(c) similar to 1.6 and the energy fluctuations diverge as similar to (eta(c) - eta)(B-1). The speed of sound decreases monotonical ly with eta and approaches the value root T/6 at the transition. (C) 1999 E lsevier Science Ltd. All rights reserved.