Molecular Theory of Irreversibility
  A generalization of the Gibbs entropy postulate is proposed based on the BBGKY hierarchy as the nonequilibrium 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 a generalization of the Liouville equation describing the evolution of the distribution vector, it is demonstrated that the entropy production is a non-negative quantity. Moreover, following the procedure of non-equilibrium thermodynamics a transport matrix is introduced and a microscopic expression for this is derived. This framework allows one to perform the thermodynamic analysis of non-equilibrium steady states which, as proven here, constitute the states of minimum entropy production when one considers small departures from stationarity. 
A. Perez-Madrid