Strong Chaos in the N-body problem and Microcanonical Thermodynamics of collisionless self gravitating systems

The dynamical justifications which lie at the basis of an effective Statistical Mechanics for self gravitating systems are formulated, analyzing some among the well known obstacles thought to prevent a rigorous Statistical treatment. It is shown that N-body gravitational systems satisfy a strong chaos criterion, so supporting the assumption of an increasingly uniform spreading of orbits over the constant energy surface, i.e., the asymptotic evolution towards a microcanonical distribution. We then focus on the necessary conditions for the equivalence of statistical ensembles and remark that this equivalence is broken for any N-body system whose interaction has a range comparable with its spatial extension. Once realized that the obstacles originate from the long range nature of the Newtonian interaction, leading to the non-extensivity of canonical and grand-canonical thermodynamic potentials, we show that instead a suitably generalized microcanonical ensemble constitutes an orthode, i.e., a reliable framework for a correct Thermodynamics. Within this setting we use the thermodynamic relations and find consistent definitions for Entropy (which turns out to be extensive, despite the non stable nature of the interaction), Temperature and Heat Capacity. Then, a Second Law-like criterion is used to select the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of self gravitating systems.