Statistical mechanics and the gravothermal catastrophe

Abstract
Jensen’s inequality is applied to the canonical partition function of a self-gravitating system to determine the best independent particle potential. The inequality allows the stability to be analyzed very easily. We recover the results of Lynden-Bell and Wood for the onset of an instability in an isothermal sphere in a heat bath. Our eigenvalue analysis leads to results very similar to those of Horwitz and Katz, but we differ in the description of the l=1 perturbation.