Strong Chaos in N-body problem and Microcanonical Thermodynamics of Collisionless Self Gravitating Systems

Abstract

An effective Microcanonical Thermodynamics of self gravitating systems(SGS) is proposed, analyzing the well known obstacles thought to prevent the formulation of a rigorous Statistical Mechanics (SM), as those due to the formal unboundedness of available phase space and to the unscreened, long range, nature of the interaction. The latter feature entails the well known inequivalence of statistical ensembles, puts clearly into question the meaning, for these systems, of the Thermodynamic Limit, and rules out the use of canonical and grand-canonical ensembles. As to the first obstacle, we argue nevertheless that a hierarchy of timescales exist such that, at any finite time, the volume of the effectively available region of phase space is indeed finite, and that the dynamics satisfies a strong chaos criterion, leading to a fast, increasingly uniform, spreading of orbits over an effectively invariant subset of the constant (N,V,E) surface; thus leading to the definition of a secularly evolving, generalized microcanonical ensemble, which allows to define an (almost extensive) effective entropy and to derive self-consistent definitions for other thermodynamic variables, giving thus an orthode for SGS. Moreover, a Second Law-like criterion allows to single out the hierarchy of secular equilibria describing, for any finite time, the macroscopic behaviour of SGS.