Generalized Gibbs Entropy, Irreversibility and Stationary States

A generalization of the Gibbs entropy postulate is proposed, based on the BBGKY hierarchy as the non-equilibrium entropy for a system of N interacting particles.This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law.By using this entropy and the methods of non-equilibrium thermodynamics in the phase space, a generalization of the Liouville equation describing the evolution of the distribution vector in the form of a master equation is obtained.After neglecting correlations in this master equation, the Boltzmann equation was obtained.Moreover, this entropy remains constant in nonequilibrium stationary states and leads to macroscopic hydrodynamics.Non-equilibrium Green-Kubo type relations and the probability for the non-equilibrium fluctuations are also derived