attractive nonintegrable large $r$ asymptotics, $U(r)\sim r^{-\a}$, $0<\a<3$, and sufficiently short-range small $r$ regularization exhibit gravitational phase transition between a relatively uniform high energy state and a low-energy state with a core-halo structure [@pr; @ki2; @ch1; @usg; @dv; @ch; @chs; @chi]. Extensive mean-field (MF) studies of the equilibrium properties of such systems [@pr; @ki2; @ch1; @usg; @dv; @ch; @chs; @chi] revealed that in a microcanonical ensemble during such a transition entropy has to undergo a discontinuous jump from a state that just ceases to be a local entropy maximum to a state with the same energy but different temperature, which is the global entropy maximum. Due to the long-range nature of gravitational interaction, the MF studies are believed to provide asymptotically (in the infinite system limit) exact information about the density and the velocity distributions and other thermodynamical parameters of the uniform state. The applicability of the MF theory to the description of the core-halo state is less obvious as the properties of a core are controlled by the short-range asymptotics of the potential.

Relatively little is known about how such a transition actually occurs, however. Youngkins and Miller [@bm2] performed a Molecular Dynamics (MD) study of a one-dimensional system 