collision integral was used by Lancellotti and Kiessling [@ki3] to prove a scaling property of the central part of the density profile. The model considered there allows the particles to escape to infinity and therefore does not have an equilibrium or even a metastable state.

There exists a vast amount of literature on cosmologically- and astrophysically- motivated studies of the temporal evolution of naturally occurring self-gravitating systems (see, e.g., Ref. [@cos] and references therein). The selection of systems and their initial and final conditions made in such studies are typically astrophysically-motivated; the considered systems are often too complex to make a general conclusions about the phase diagrams and phase transitions in such systems.

In this paper we present MD studies of gravitational collapse, and reverse to collapse, i.e., explosion, transitions in a microcanonical ensemble of self-attracting particles. Besides their pure statistical mechanical implications, these studies represent our attempt to bridge a gap between the usually complicated MD and hydrodynamic simulations of the realistic astrophysical systems and the MF analysis of the phase diagram of simple self-gravitating models.

A system with soft Coulomb potential $-(r^2+r_0^2)^{-1/2}$ is considered. Such systems are have been studied using both MF theory (see, e.g., Refs. [@usg; 