The Second Law and Boltzmann'sH-Theorem

The kinetic and statistical explanations of the second law of thermodynamics is examined in the region of extended irreversible thermodynamics. The thermodynamical entropy is identified with the entropy obtained by the maximum entropy procedure with constraints. The differential of the thermodynamical entropy is exact in the Gibbs space of fundamental variables. It is shown that irreversibility in mesoscopic levels is necessary but not sufficient for the entropy to increase. Here, irreversibility is expressed in terms of the Boltzmann equation or in terms of the Markovian master equation of the dynamical semi-group theory. The difference between the H -theorem and the second law is clearly demonstrated in the extended Gibbs space. In addition, the relative entropy is considered as an entity to connect the H -theorem with the second law.