Inconsistency in theories of violent relaxation

We examine an inconsistency in theories of violent relaxation by Lynden-Bell and Nakamura. The inconsistency arises from the non-transitive nature of these theories: a system that undergoes a violent relaxation, relaxes and then, upon an addition of energy, undergoes violent relaxation once again would settle in an equilibrium state that is different from the one that is predicted had the system gone directly from the initial to the final state. We conclude that a proper description of the violent relaxation process cannot be achieved by an equilibrium statistical mechanics approach, but instead a dynamical theory for the coarse-grained phase-space density is needed.