Collisionless relaxation in gravitational systems: From violent relaxation to gravothermal collapse

Theory and simulations are used to study collisionless relaxation of a gravitational $N$-body system. It is shown that when the initial one particle distribution function satisfies the virial condition -- potential energy is minus twice the kinetic energy -- the system quickly relaxes to a metastable state described {\it quantitatively} by the Lynden-Bell distribution with a cutoff. If the initial distribution function does not meet the virial requirement, the system undergoes violent oscillations, resulting in a partial evaporation of mass. The leftover particles phase separate into a core-halo structure. The theory presented allows us to quantitatively predict the amount and the distribution of mass left in the central core, without any adjustable parameters. On a longer time scale $\tau_G \sim N$ collisionless relaxation leads to a gravothermal collapse.