Statistical theory of violent relaxation

The final state of violent relaxation is discussed. We have made N-body simulations in a one-dimensional sheet model and analyzed the evolution of the system from the standpoint of coarse-grained entropy. It increases in the initial phase of evolution. However, we find that it does not necessarily reach the maximum value for the given values of the total mass, the total energy, and the fine-grained phase space density despite the Lynden-Bell theory. The difference in the coarse-grained entropies between the final structure thus obtained and the maximum value is so large that the appearance of such a structure seems impossible. This means that the violent relaxation does not lead to the ergodicity. Discrepancies between the theory and previous numerical simulations are also understood naturally from this result. (author)