Phase diagram of self-attracting systems

Phase diagram of microcanonical ensembles of self-attracting particles is studied for two types of short-range potential regularizations: self-gravitating fermions and classical particles interacting via attractive soft $-(r^2+r_0^2)^{-1/2}$ Coulomb potential. When the range of regularization is sufficiently short, the self-attracting systems exhibit gravitational or collapse-like transition. As the fermionic degeneracy or the softness radius increases, the gravitational phase transition crosses over to a normal first-order phase transition, becomes second-order at a critical point, and finally disappears. Applicability of a commonly used saddle-point or mean-field approximation and importance of metastable states is discussed.