  The role of thermodynamics in the evolution of systems evolving under purely
gravitational forces is not completely established. Both the infinite range and
singularity in the Newtonian force law preclude the use of standard techniques.
However, astronomical observations of globular clusters suggests that they may
exist in distinct thermodynamic phases. Here, using dynamical simulation, we
investigate a model gravitational system which exhibits a phase transition in
the mean field limit. The system consists of rotating, concentric, mass shells
of fixed angular momentum magnitude and shares identical equilibrium properties
with a three dimensional point mass system satisfying the same condition. The
mean field results show that a global entropy maximum exists for the model, and
a first order phase transition takes place between "quasi-uniform" and
''core-halo'' states, in both the microcanonical and canonical ensembles. Here
we investigate the evolution and, with time averaging, the equilibrium
properties of the isolated system. Simulations were carried out in the
transition region, at the critical point, and in each clearly defined
thermodynamic phase, and striking differences were found in each case. We find
full agreement with mean field theory when finite size scaling is accounted
for. In addition, we find that (1) equilibration obeys power law behavior, (2)
virialization, equilibration, and the decay of correlations in both position
and time, are very slow in the transition region, suggesting the system is also
spending time in the metastable phase, (3) there is strong evidence of
long-lived, collective, oscillations in the supercritical region.
