Title: The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation,
Abstract: 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 potential. It
is traditionally assumed that this evolution is governed by the Vlasov-Poisson
equation, in which case entropy must be conserved. We run N-body simulations of
isolated self-gravitating systems, using three simulation 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. At each snapshot, we estimate the Shannon entropy of the
distribution function with three different techniques: Kernel, Nearest Neighbor
and EnBiD. For all simulation codes and estimators, the entropy evolution
converges to the same limit as N increases. During violent relaxation, the
entropy has a fast increase followed by damping oscillations, indicating that
violent relaxation must be described by a kinetic equation other than the
Vlasov-Poisson, even for N as large as that of astronomical structures. This
indicates that violent relaxation cannot be described by a time-reversible
equation, shedding some light on the so-called "fundamental paradox of stellar
dynamics". The long-term evolution is well described by the orbit-averaged
Fokker-Planck model, with Coulomb logarithm values in the expected range 10-12.
By means of NBODY-2, we also study the dependence of the 2-body relaxation
time-scale on the softening length. The approach presented in the current work
can potentially provide a general method for testing any kinetic equation
intended to describe the macroscopic evolution of N-body systems.