Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems confined in a finite domain consisting of either point masses or rotating mass shells of different dimension. We establish a direct connection between the spherically symmetric equilibrium states of a self-gravitating point mass system and a shell model of dimension 3. We construct the equilibrium density functions by maximizing the entropy subject to the usual constraints of normalization and energy, but we also take into account the constraint on the sum of the squares of the individual angular momenta, which is also an integral of motion for these symmetric systems. Two statistical ensembles are introduced that incorporate the additional constraint. They are used to investigate the possible occurrence of a phase transition as the defining parameters for each ensemble are altered.

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.

The role of thermodynamics in the evolution of systems evolving under purely gravitational forces is not completely established. Both the infinite range and singularity in the Newtonian force law preclude the use of standard techniques. However, astronomical observations of globular clusters suggests that they may exist in distinct thermodynamic phases. Here, using dynamical simulation, we investigate a model gravitational system which exhibits a phase transition in the mean field limit.The system consists of rotating, concentric, mass shells of fixed angular momentum magnitude and shares identical equilibrium properties with a three dimensional point mass system satisfying the same condition. The mean field results show that a global entropy maximum exists for the model, and a first order phase transition takes place between "quasi-uniform" and "core-halo" states, in both the microcanonical and canonical ensembles. Here we investigate the evolution and, with time averaging, the equilibrium properties of the isolated system. Simulations were carried out in the transition region, at the critical point, and in each clearly defined thermodynamic phase, and striking differences were found in each case. We find full agreement with mean field theory when finite size scaling is accounted for. In addition, we find that (1) equilibration obeys power law behavior, (2) virialization, equilibration, and the decay of correlations in both position and time, are very slow in the transition region, suggesting the system is also spending time in the metastable phase, (3) there is strong evidence of long-lived, collective, oscillations in the supercritical region.

We investigate the dynamical evolution of gravothermal catastrophe in a model of a spherical cluster where, besides the energy and angular momentum, an additional integral of motion is also taken into account. Using dynamical simulation, we study a system of concentric, rotating, spherical shells employing a precise, event-driven, algorithm that permits the controlled exchange of internal angular momentum. Initially the system starts to relax to a locally stable state that is in good agreement with mean field predictions. This is followed by core collapse with the development of a core-halo structure and gravothermal oscillation.

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