Statistical mechanics and phase diagrams of rotating self-gravitating fermions

Abstract

We compute statistical equilibrium states of rotating self-gravitating fermions by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. We describe the phase transition from a gaseous phase to a condensed phase (corresponding to white dwarfs, neutron stars or fermion balls in dark matter models) as we vary energy and temperature. 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 (or system size) and angular velocity. We argue that systems described by the Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric structures when rotation is increased, in continuity with the case of polytropes with index n > 0.808 (the Fermi gas at T = 0 corresponds to n = 3/2). This differs from the study of Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration space appropriate to stellar formation and find "double star" structures (their model at T = 0 corresponds to n = 0). We also consider the case of classical objects described by the Boltzmann entropy and discuss the influence of rotation on the onset of gravothermal catastrophe (for globular clusters) and isothermal collapse (for molecular clouds). 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 (MCE) and Dirac peaks (or massive condensates) in the canonical ensemble (CE). We also describe an hysteretic cycle between the gaseous phase and the condensed phase that are connected by a "collapse" or an "explosion".