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The collapse of a collisionless self-gravitating system, with the fastachievement of a quasi-stationary state, is driven by violent relaxation, witha typical particle interacting with the time-changing collective potential. Itis traditionally assumed that this evolution is governed by the Vlasov-Poissonequation, in which case entropy must be conserved. We run N-body simulations ofisolated 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 andinitial conditions. At each snapshot, we estimate the Shannon entropy of thedistribution function with three different techniques: Kernel, Nearest Neighborand EnBiD. For all simulation codes and estimators, the entropy evolutionconverges to the same limit as N increases. During violent relaxation, theentropy has a fast increase followed by damping oscillations, indicating thatviolent relaxation must be described by a kinetic equation other than theVlasov-Poisson, even for N as large as that of astronomical structures. Thisindicates that violent relaxation cannot be described by a time-reversibleequation, shedding some light on the so-called "fundamental paradox of stellardynamics". The long-term evolution is well described by the orbit-averagedFokker-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 relaxationtime-scale on the softening length. The approach presented in the current workcan potentially provide a general method for testing any kinetic equationintended to describe the macroscopic evolution of N-body systems.

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