Statistical mechanics and phase diagrams of rotating self-gravitating fermions

We compute statistical equilibrium states of rotating self-gravitating systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. The Fermi-Dirac distribution describes quantum particles (fermions) subject to Pauli’s exclusion principle. It is also a typical prediction of Lynden-Bell’s theory of violent relaxation for collisionless stellar systems (in that case degeneracy accounts for the Liouville theorem). We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter and angular velocity. We argue that systems described by the Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric structures, in continuity with the case of polytropes with index n > 0.808 (the Fermi gas at T = 0 corresponds to n = 3/2). This contrasts with the study of Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration space and find “double star” structures (their model at T = 0 corresponds to n = 0). We also discuss the influence of rotation on the onset of the gravothermal catastrophe for globular clusters. On general grounds, we complete previous investigations concerning the nature of phase transitions in self-gravitating systems. We emphasize the inequivalence of statistical ensembles regarding the formation of binaries (or low-mass condensates) in the microcanonical ensemble and Dirac peaks (or massive condensates) in the canonical ensemble. We also describe an hysteretic cycle between the gaseous phase and the condensed phase that are connected by a “collapse” or an “explosion”.