In a physical system consisting of a collection of particles, information can be obtained from the way in which the particles are distributed in space. For a Poissonian distribution in equilibrium there is no correlation among the particles, the position of each one being independent of the position of the others. This distribution sets the particles randomly in space with equal probability for all the positions. In contrast, when the distribution deviates from Poissonian, the correlations among particles are not null; instead, the particles will be mutually interacting.
The relationship between the correlations of the distribution and the interaction force is called the force equation and can be obtained under a certain hypothesis (also called the cosmic virial theorem in the literature relating to large-scale structure; Peebles 1976). The development of statistical-mechanical theory of liquids has allowed this relationship to be achieved in the specific case where the particles are all equal, point-like and in Boltzmann equilibrium, which is quite common for classical particles in thermodynamic equilibrium (see for example Goodstein 1975, March & Tosi 1976). In general, statistical-mechanical tools are applied to obtain the distribution of particles (molecules on the atomic scale) from an assumed form of the interaction (the Lennard-Jones interaction among molecules).
Since these statistical applications can be developed in environments other than on the atomic scale, my purpose is to derive a few relationships that will be useful in cosmology. Groups of galaxies in the large-scale structure of the Universe are observed in sky surveys, i.e. we have a distribution of particles in space although there are some differences with respect to the liquid. First of all, we know that these particles are not all identical; their mass and other characteristics that are not important dynamically are distinctive among themselves. In view of this, we could face problems when applying statistical mechanics on astronomical scales, but the issue of different masses is still an important difficulty to resolve first.
Our purpose is to obtain a force equation for a distribution of particles with different masses that will allow different magnitudes of interest to be related among themselves. Our result will apply directly to Newtonian gravity for a Universe in expansion.
The necessary condition for solving the problem is to derive information about the probabilities of different configurations. The Maxwell-Boltzmann equilibrium for classical particles is the case that I will solve. This approach was already considered by other authors for the largest scales in the Universe (e.g. Saslaw & Hamilton 1984; Saslaw 1985) and I also believe that this condition could be applicable in some gravitational systems and in some scales (not all though, see Betancort-Rijo 1988): for example, the distribution of galaxies may be a good candidate to consider relaxed in which we will develop an example of the expression derived here in Sect. VIII. Nevertheless, there might be other gravitational systems for which this approach may be valid.
© European Southern Observatory (ESO) 1997
Online publication: October 15, 1997