Approach to equilibrium in $N$-body gravitational systems

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 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 center 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. This last result, along with the observation of the anomalously rapid mass segregation in some situations, suggests that, in gravitational systems, different quantities may relax at different rates, and that the thermal (two body) relaxation time scale, even if accurate for energy relaxation of single mass systems, may not be universal. This in turn indicates that the issue of relaxation in gravitational systems is far from being a closed subject.