Lpthe-96/13; Demirm-96 Fractal Dimensions and Scaling Laws in the Interstellar Medium: a New Field Theory Approach

We develop a field theoretical approach to the cold interstellar medium (ISM). We show that a non-relativistic self-gravitating gas in thermal equilibrium with variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field φ(~x) with exponential self-interaction. We analyze this field theory perturbatively and non-perturbatively through the renormalization group approach. We show scaling behaviour (critical) for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation ∆M(R) ∼ RdH for the mass on a region of size R, and ∆v ∼ Rq for the velocity dispersion where q = 12(dH − 1). For the density-density correlations we find a power-law behaviour for large distances ∼ |~r1 − ~r2| 2dH−6. The fractal dimension dH turns to be related with the critical exponent ν of the correlation length by dH = 1/ν. The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behaviour is governed by the (non-perturbative) Ising fixed point. The corresponding values of the scaling exponents are ν = 0.631..., dH = 1.585... and q = 0.293.... Mean field theory yields for the scaling exponents ν = 1/2, dH = 2 and q = 1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4 ≤ dH ≤ 2, 0.3 ≤ q ≤ 0.6 . As typical in critical phenomena, the scaling behaviour and critical exponents of the ISM can be obtained without dwelling into the dynamical (time-dependent) behaviour. The relevant rôle of selfgravity is stressed by the authors in a Letter to Nature, September 5, 1996. 98.38.-j, 11.10.Hi, 05.70.Jk Typeset using REVTEX