The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation

Research paper by Leandro Beraldo e Silva, Walter de Siqueira Pedra, Laerte Sodré, Eder Perico, Marcos Lima

Indexed on: 21 Mar '17Published on: 21 Mar '17Published in: arXiv - Astrophysics - Astrophysics of Galaxies


The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective gravitational potential. It is traditionally assumed that this evolution is described by the (time-reversible) Vlasov-Poisson equation, in which case entropy must be conserved. We use N-body simulations to follow the evolution of an isolated self-gravitating system, estimating the (fine-grained) distribution function and the corresponding Shannon entropy. We do this with three different codes: NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening) and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. We find that during violent relaxation entropy increases in a way that cannot be described by 2-body relaxation as modeled by the Fokker-Planck approximation. On the other hand, the long-term evolution is very well described by this model. Our results imply that the violent relaxation process must be described by a kinetic equation other than the Vlasov-Poisson, even if the system is collisionless. Our estimators provide a general method for testing any proposed kinetic equation. We also study the dependence of the 2-body relaxation time-scale $\tau_{col}$ on the number of particles N, obtaining $\tau_{col}\propto \sqrt{N}$, and the dependence of $\tau_{col}$ on the softening length $\varepsilon$, which can be fit by a function of the form $\tau_{col} \propto \sqrt{\varepsilon}\cdot e^{c\varepsilon}$, for a fixed number of particles.