Phase transitions in self-gravitating systems

We discuss the nature of phase transitions in self-gravitating systems. We show the connection between the binary star model of Padmanabhan, the thermodynamics of stellar systems and the thermodynamics of self-gravitating fermions. We stress the inequivalence of statistical ensembles for systems with long-range interactions, like gravity. In particular, we contrast the microcanonical evolution of stellar systems from the canonical evolution of self-gravitating Brownian particles. At low energies, self-gravitating Hamiltonian systems experience a gravothermal catastrophe in the microcanonical ensemble. At low temperatures, self-gravitating Brownian systems experience an isothermal collapse in the canonical ensemble. For classical particles, the gravothermal catastrophe leads to a binary star surrounded by a hot halo while the isothermal collapse leads to a Dirac peak containing all the mass. For self-gravitating fermions, the collapse stops when quantum degeneracy comes into play through the Pauli exclusion principle. The end-product of the collapse is a fermion ball, resembling a cold white dwarf star, surrounded by a halo. We can thus describe a phase transition from a gaseous phase to a condensed phase. At high energies or high temperatures, the condensate can experience an explosion, reverse to the collapse, and return to the gaseous phase. Due to the existence of long-lived metastable states, the points of collapse and explosion differ. This leads to a notion of hysteretic cycle in microcanonical and canonical ensembles.