Models of dark matter halos based on statistical mechanics: II. The
  fermionic King model
  We discuss the nature of phase transitions in the fermionic King model which
describes tidally truncated quantum self-gravitating systems. This distribution
function takes into account the escape of high energy particles and has a
finite mass. On the other hand, the Pauli exclusion principle puts an upper
bound on the phase space density of the system and stabilizes it against
gravitational collapse. As a result, there exists a statistical equilibrium
state for any accessible values of energy and temperature. We plot the caloric
curves and investigate the nature of phase transitions as a function of the
degeneracy parameter in both microcanonical and canonical ensembles. We
consider stable and metastable states and emphasize the importance of the
latter for systems with long-range interactions. Phase transitions can take
place between a "gaseous" phase unaffected by quantum mechanics and a
"condensed" phase dominated by quantum mechanics. The phase diagram exhibits
two critical points, one in each ensemble, beyond which the phase transitions
disappear. There also exist a region of negative specific heats and a situation
of ensemble inequivalence for sufficiently large systems. We apply the
fermionic King model to the case of dark matter halos made of massive
neutrinos. The gaseous phase describes large halos and the condensed phase
describes dwarf halos. Partially degenerate configurations describe
intermediate size halos. We argue that large dark matter halos cannot harbor a
fermion ball because these nucleus-halo configurations are thermodynamically
unstable (saddle points of entropy). Large dark matter halos may rather contain
a central black hole resulting from a dynamical instability of relativistic
origin occurring during the gravothermal catastrophe.
