Classical Partition Function for Non-Relativistic Gravity

Mir Hameeda,Angelo Plastino,Javier Zamora,Mario Carlos Rocca

doi:10.3390/axioms10020121

Abstract

We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution e−βHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, it was shown in (Physica A 497 (2018) 310), by appeal to sophisticated mathematics developed in the second half of the last century, that this is not so. Z can indeed be computed by recourse to (A) the analytical extension treatments of Gradshteyn and Rizhik and Guelfand and Shilov, that permit tackling some divergent integrals and (B) the dimensional regularization approach. Only one special instance was discussed in the above reference. In this work, we obtain the classical partition function for Newton’s gravity in the four cases that immediately come to mind.

Highlights

This paper is a continuation and generalization of [1]
An important point that will emerge below is that of the behavior of the partition function Z as a function of the inverse temperature β
In this work, we showed how to deal with the partition function for gravitational systems in four different scenarios:

Summary

Introduction

It involves mathematical ideas that were fully explored there (and references therein), in which a canonical ensemble at the temperature T was concocted for a two-particle gravitation system and fully analyzed. It is advisable to have [1] at hand in trying to follow our discussion below. A very important concept is that of generalized dimensionally regularized partition function Z, which we abbreviate as GDR[Z. An important point that will emerge below is that of the behavior of the partition function Z as a function of the inverse temperature β. (1) As gravitational binding gets tighter, the kinetic energy augments and so does, as a consequence, the temperature;. (2) As gravitational binding gets weaker, the kinetic energy decreases, and as a consequence, the temperature diminishes;. (3) The specific heat becomes negative if the system can freely expand or contract

Basic Instance

Conclusions