Statistical Mechanics of Infinite Gravitating Systems

Abstract

The cosmological many‐body problem was stated over 300 years ago, but its solution is quite recent and still incomplete. Imagine an infinite expanding universe essentially containing a very large number of objects moving in response to their mutual gravitational forces. What will be the spatial and velocity distributions of these objects and how will they evolve? This question fascinates on many levels. Though inherently non‐linear, it turns out to be one of the few analytically solvable problems of statistical mechanics with long range forces. The partition function can be calculated. From this all the thermodynamic properties of the system can be obtained for the grand canonical ensemble. They confirm results derived independently directly from the first and second laws of thermodynamics. The behavior of infinite gravitating systems is quite different from their finite relations such as star clusters. Infinite gravitating systems have regimes of negative specific heat, an unusual type of phase transition, and a very close relation to the observed large‐scale structure of our universe. This last feature provides an additional astronomical motivation, especially since the statistical mechanics may be generalized to include effects of dark matter haloes around galaxies. Previously the cosmological many‐body problem has mostly been studied using the BBGKY hierarchy (not so suitable in the non‐linear regime) and by direct computer integrations of the objects' orbits. The statistical mechanics agrees with and substantially extends these earlier results. Most astrophysicists had previously thought that a statistical thermodynamic approach would not be applicable because: a) many‐body gravitational systems have no rigorous equilibrium state, b) the unshielded nature of the long‐range force would cause the partition function to diverge on large scales, and c) point masses would produce divergences on small scales. However, deeper considerations show that these are not significant difficulties. There remain many important questions for which we have only a very preliminary understanding. An example is to find the basin of attraction for initial states which are able to evolve into systems described by this gravitational partition function. It may well be that ideas and techniques developed for other types of statistical mechanical systems will help us answer these questions.