Dimensionally regularized Boltzmann-Gibbs Statistical Mechanics and two-body Newton's gravitation

Abstract

It is believed that the canonical gravitational partition function $Z$ associated to the classical Boltzmann-Gibbs (BG) distribution $\frac {e^{-\beta H}} {{\cal Z}}$ cannot be constructed because the integral needed for building up $Z$ includes an exponential and thus diverges at the origin. We show here that, by recourse to 1) the analytical extension treatment obtained for the first time ever, by Gradshteyn and Rizhik, via an appropriate formula for such case and 2) the dimensional regularization approach of Bollini and Giambiagi's (DR), one can indeed obtain finite gravitational results employing the BG distribution. The BG treatment is considerably more involved than its Tsallis counterpart. The latter needs only dimensional regularization, the former requires, in addition, analytical extension.