Fractal Dimensions and Scaling Laws in the Interstellar Medium: a new Field Theory approach

Abstract

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({\vec 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 Delta M(R) \sim R^{d_H} for the mass on a region of size R, and Delta v \sim R^q for the velocity dispersion where q = (d_H -1)/2. For the density-density correlations we find a power-law behaviour for large distances \sim|{\vec r_1} -{\vec r_2}|^{2 d_H -6}. The fractal dimension d_H turns to be related with the critical exponent nu of the correlation length by d_H = 1/nu. 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 nu = 0.631..., d_H = 1.585... and q = 0.293.... Mean field theory yields for the scaling exponents nu =1/2, d_H = 2 and q = 1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4 \leq d_H \leq 2, 0.3 \leq q \leq 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 self- gravity is stressed by the authors in a Letter to Nature, September 5, 1996.

Latex file, 25 pages, no figures. Few minor changes. To appear in Phys. Rev. D, November 15, 1996

References in corpus (6)

• Derivative Expansion of the Exact Renormalization Group
• On Truncations of the Exact Renormalization Group
• On the fractal structure of two-dimensional quantum gravity
• Scaling in quantum gravity
• Is Dark Matter in Spiral Galaxies Cold Gas? I. Observational Constraints and Dynamical Clues About Galaxy Evolution
• Is Dark Matter in Spiral Galaxies Cold Gas? II. Fractal Models and Star Non-Formation

Cited by in corpus (29)

• Control of star formation by supersonic turbulence
• Gravitational Collapse in Turbulent Molecular Clouds. I. Gasdynamical Turbulence
• Gravitational instability of finite isothermal spheres
• Statistical Mechanics of the Self-Gravitating Gas: I. Thermodynamic Limit and Phase Diagram
• Coarse-Grained Effective Action and Renormalization Group Theory in Semiclassical Gravity and Cosmology
• Origin of scaling structure and non-gaussian velocity distribution in self-gravitating ring model
• Gravitational instability of finite isothermal spheres in general relativity. Analogy with neutron stars
• Statistical Mechanics of the Self-Gravitating Gas: II. Local Physical Magnitudes and Fractal Structures
• The fractal structure of the universe : a new field theory approach
• Self-gravitating fluid dynamics, unstabilities and solitons
• Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas
• Density distribution function of a self-gravitating isothermal compressible turbulent fluid in the context of Molecular Clouds ensembles
• Renormalization Group Flow and Fragmentation in the Self-Gravitating Thermal Gas
• Caloric curves of classical self-gravitating systems in general relativity
• Phase Transition in a Self--Gravitating Planar Gas
• Field Theory of the Correlation Function of Mass Density Fluctuations for Self-Gravitating Systems
• Statistical Mechanics of the Self-gravitating gas with two or more kinds of Particles
• Field Equation of Correlation Function of Mass Density Fluctuation for Self-Gravitating Systems
• Field Equation of Correlation Function of Mass Density Fluctuation for Self-Gravitating Systems
• Statistical Mechanics of the Self-Gravitating Gas: Thermodynamic Limit, Unstabilities and Phase Diagrams
• Relative chaos in gravitating systems with massive centre
• Scale Symmetry in the Universe
• Fractal structures and the large scale distribution of galaxies
• Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: gaseous density profiles and onset of condensation
• Complexity in Cosmology
• Phase structure of self-gravitating systems
• Phase transition in Modified Newtonian Dynamics (MONDian) self-gravitating systems
• Cluster formation in the Fermi system with long-range interaction
• The nonlinear equation of correlation function of galaxies in the expanding universe and the solution in linear approximation