Boltzmann conjecture, meta-equilibrium entropy, second law, chaos and irreversibility for many body systems

A heuristic generalization of the Boltzmann-Gibbs microcanonical entropy is proposed, able to describe meta-equilibrium features and evolution of macroscopic systems. Despite its simple-minded derivation, such a function of "collective parameters" characterizing the microscopic state of N-body systems, yields, at one time, a statistical interpretation of dynamic evolution, and dynamic insights on the basic assumption of statistical mechanics. Its natural (implicit) time dependence entails} a "Second Law-like" behaviour and allows moreover, to perform an elementary test of the Loschmidt reversibility objection, pointing out the crucial relevance of Chaos in setting up effective (statistico-mechanical and dynamical) "arrows of time". Several concrete (analytical and numerical) applications illustrate its properties.