  Collapse, or a gravitational-like phase transition is studied in a
microcanonical ensemble of particles with an attractive $1/r^{\alpha}$
potential. A mean field continuous integral equation is used to determine a
saddle-point density profile that extremizes the entropy functional. For all
$0<\alpha<3$, a critical energy is determined below which the entropy of the
system exhibits a discontinuous jump. If an effective short-range cutoff is
applied, the entropy jump is finite; if not, the entropy diverges to $+\infty$.
A stable integral equation solution represents a state with maximal entropy;
the reverse is always true only for a modified integral equation introduced
here.

Statistical Mechanics;Astrophysics