Abstract We investigated how a gravitational collisionless system approaches to equilibrium through so-called violent relaxation. A system which is far from dynamical equilibrium approaches to dynamical equilibrium through the dissipation of energy from the coherent motion of particles to random motion by wave-particle interaction and phase mixing. Particles change their energies by the wave-particle interaction with the coherent motion of particles. We performed direct N-body simulations of collisionless systems and found (a) that the direction of the evolution through the wave-particle interaction is different from that due to the entropy maximum, and (b) that wave-particle coupling disappears within a few crossing times. Thus, violent relaxation does not lead the system to a thermally relaxed state. In other words, it is not a relaxation process. Evolution through the wave-particle interaction does not lead to thermal equilibrium, since it tries to establish an equipartition between the bulk motion and that of individual particles. As a result, all of the particles are heated, regardless of their initial energies. The wave, itself, decays quickly, since it loses energy through this wave-particle interaction, itself. Within a few crossing times, the wave becomes sufficiently small that the wave-particle interaction has a negligible effect on to the energy of the particles. At this stage, however, the system is still not in an dynamical equilibrium. We found that even in this stage the central density shows an oscillating behavior with an amplitude larger than 10%. This oscillation can continue for ~ 10 crossing times or longer. This long lifetime is explained by the fact that it decays only through phase mixing. This phase mixing is slow in the core, since particles have very similar orbital periods there.