  The evolution of closed gravitational systems is studied by means of $N$-body
simulations. This, as well as being interesting in its own right, provides
insight into the dynamical and statistical mechanical properties of
gravitational systems: the possibility of the existence of stable equilibrium
states and the associated relaxation time would provide an ideal situation
where relaxation theory can be tested. Indeed, these states are found to exist
for single mass $N$-body systems, and the condition condition for this is
simply that obtained from elementary thermodynamical considerations applied to
self-gravitating ideal gas spheres. However, even when this condition is
satisfied, some initial states may not end as isothermal spheres. It is
therefore only a necessary condition. Simple considerations also predict that,
for fixed total mass, energy and radius, stable isothermal spheres are unique.
Therefore, statistically irreversible perturbations to the density profile
caused by the accumulation of massive particles near the centre of multimass
systems, destroy these equilibria if the aforementioned quantities are kept
fixed. The time-scale for this to happen was found to be remarkably short (a
few dynamical times when $N= 2500$) in systems undergoing violent relaxation.
The time taken to achieve thermal equilibrium depended on the initial
conditions and could be comparable to a dynamical time (even when the
conditions for violent relaxation were not satisfied) or the two body
relaxation time. The relaxation time for velocity anisotropies was intermediate
between these two time-scales, being long compared to the dynamical time but
much (about four times) shorter than the time-scale of energy relaxation.
