Relative Entropies and Irreversible Thermodynamics

A critical discussion is given of the idea that the relation between microscopic dynamics and thermodynamics can be based on the concept of relative entropy. It is shown how dissipation can be treated in the response theory under certain conditions. A thermodynamic interpretation of the dynamical semigroup theory for a system interacting with its surroundings is given. By utilizing the method of maximum entropy with the well specified set of relevant observables, the construction of thermodynamic entropy is demonstrated. It is argued that the statistical (say, information) entropy cannot be identified with the thermodynamic entropy and that the relation between the two entropies is written in terms of the relative entropy. In this form, the distinction between the H-theorem and the second law of thermodynamics is easily made. It should be kept in mind that the H-theorem is for the kinetic stage while the second law is for the thermodynamic (or, hydrodynamic) stage.