Rotation-induced phase transition in a spherical gravitating system

Due to the infinite range and singularity of the gravitational force, it is difficult to directly apply the standard methods of statistical physics to self-gravitating systems, e.g., interstellar grains, globular clusters, galaxies, etc. Unusual phenomena can occur, such as a negative heat capacity, unbounded mass, or the gravothermal catastrophe where the equilibrium state is fully collapsed and the entropy is unbounded. Using mean field theory, we investigate the influence of rotation on a purely spherical gravitational system. Although spherical symmetry nullifies the total angular momentum, its square is finite and conserved. Here we study the case where each particle has specific angular momentum of the same magnitude l. We rigorously prove the existence of an upper bound on the entropy and a lower bound for the energy. We demonstrate that, in the microcanonical and canonical ensembles, a phase transition occurs when l falls below a critical value. We characterize the properties of each phase and construct the coexistence curve for each ensemble. Possible applications to astrophysics are considered.