Press-schechter, Thermodynamics, and Gravitational Clustering

For the special case of nonlinear gravitational clustering from an initially Poisson distribution , the counts-in-cells distribution function is obtained from the excursion set, Press-Schechter multiplicity function. This Poisson Press-Schechter distribution function has the same form as the gravitational quasi-equilibrium counts-in-cells distribution function, predicted by the Saslaw-Hamilton thermodynamic model of nonlinear gravi-tational clustering, that ts the observed galaxy distribution well. By changing an ad hoc guess in the Saslaw-Hamilton thermodynamic model, the Negative Binomial distribution (which also ts relevant observations well) is derived from the thermodynamic approach. At present, analytic simplicity is the primary reason for preferring one guess over another, so the thermodynamic approach does not, at present, yield a unique prediction for the gravitational counts-in-cells distribution function. Two ways to constrain the parameter space of possible guesses are described; one of these suggests that the Negative Binomial is not a physically reasonable model. One possible relation between the original Saslaw-Hamilton thermodynamic model and the Poisson Press-Schechter approaches is obtained. A system of non-interacting, virialized clusters having a range of masses, the distribution of masses being given by the Poisson Press-Schechter multiplicity function, is shown to be consistent with the original Saslaw-Hamilton thermodynamic model. For this model to work the virialized clusters must be in thermal equilibrium with each other, so that all clusters have the same temperature, independent of their masses. This last requirement, and the idealization that the clusters do not interact gravitationally with each other, are in contradiction with observations.