Time evolution of entropy for spherical self-gravitating systems

In this work, we investigate the dynamical evolution of spherical self-gravitating systems under their own gravity with N-body simulations. For this purpose, we study the evolution of the generalized virialization relations, and particularly focus on the time evolution of the coarse-grained entropy of dark matter halos under various perturbations. First, we construct six single perturbation models under four initial conditions to mimic typical disturbances that a realistic gravitating system may encounter. With the simulation results, we show the time evolution of the entropy for the six perturbation models. In all these models, at first the entropy increases rapidly for a short period of time, slowly evolves for a longer period of time and then remains nearly unchanged in the subsequent evolution. The main dynamical mechanisms behind these evolutions should be violent relaxation and phase mixing. However, under repeated perturbations to the system, the evolution of entropy of self-gravitating systems manifests complete differences from that of the usual thermodynamical systems. We see that the entropy of the end states of every single perturbation, according to different repeated perturbation modes, either decreases or increases. We argue that the increasing or decreasing of the end-state entropy should be the reflection of the complexity of the thermodynamical states of self-gravitating systems. These conclusions are independent of the initial conditions. Besides, we demonstrate that the generalized virialization relations can reveal whether or not, or in which radius interval, the collisionless Boltzmann equation is suitable for description of a self-gravitating system, and can be used as good stability criteria of the system.