Statistical Mechanics of planar stellar systems: Solving divergences in self-gravitational systems

It is believed that the canonical gravitational partition function associated with the two-body interacting Newton’s gravitation cannot be constructed because the concomitant integral is exponentially divergent. We showed previously that one can indeed obtain finite gravitational results employing both the Gibbs–Boltzmann distribution and Tsallis’ one, by recourse to the analytical extension treatment and the generalization of Bollini and Giambiagi’s dimensional regularization. We deal here with a model of disc galaxy with a supermassive black hole at its center. Some interesting and coherent results emerge: i—an upper bound in the temperature, ii—the specific heat is negative, iii—the limit of the specific heat when the mass of the black-hole tends to zero is −kB, iv—the third law of thermodynamics is violated, and v—the gravothermal catastrophe is avoided if the number of constituents of a surrounding halo is equal or less than the number of stars in the galaxy.