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B. Statistical mechanics

The statistical mechanics of a self-gravitating system is a totally nontrivial subject. Most of the difficulty arises from the fact that gravitation is an always-attractive force of infinite range: there is no analogue to the Debye shielding in plasma physics. Perhaps the most outstanding success was the discovery by Jeans in the 1920's of equilibrium solutions to the Liouville equation for the distribution function of a collection of stars (the Jeans Theorem). This has led to a whole industry in galaxy dynamics, but it has had little or no impact on cosmology where we might like to view the expanding universe with galaxies condensing out as a phase transition in action.

This has not deterred the brave from tackling the statistical mechanics or thermodynamics of self-gravitating systems, but it is perhaps fair to say that so far there have been few outstanding successes. The discussion by Lynden-Bell and Wood (1968) of the so-called gravo-thermal collapse of a stellar system in a box is probably as close as anyone has come. It was only in the 1970's that cosmologists "discovered" the two-point clustering correlation function for the distribution of galaxies and it was not until the late 1980's with the discovery by de Lapparent et al. (1986) of remarkable large scale cosmic structure that we even knew what it was we were trying to describe.

The early work of Saslaw (1968, 1969) on "Gravithermodynamics" predated the knowledge of the correlation function. Following the discovery of the correlation function we saw the work of Fall and Severne (1976), Kandrup (1982), and Fry (1984b), providing models for the evolution of the correlation function in various approximations.

One major problem was how to describe this structure. By 1980, it was known that the two-point correlation function looked like a power law on scales 1 < 10h-1 Mpc. It was also known that the 3-point function too had a power law behavior and that it was directly related to sums of products of pairs of two-point functions (rather like the Kirkwood approximation). However, N-point correlation functions were not really evocative of the observed structure and were difficult to measure past N = 4.

Two suggestions for describing large scale cosmic structure emerged: void probability functions proposed by White (1979) and measured first by Maurogordato and Lachieze-Rey (1987) and multifractal measures (Jones et al., 1988), the latter being largely motivated by the manifest scaling behavior of the lower order correlation functions on scales < 10h-1 Mpc. Both of these descriptors encapsulate the behavior of high order correlation functions.

1 The natural unit of length to describe the large scale structure is the megaparsec (Mpc): 1 Mpc = 106 pc appeq 3.086 × 1022 m appeq 3.26 × 106 light years. h is the Hubble constant in units of 100 Mpc-1 km s-1. Back.

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