Entropy for generalised quantum kinetic and Liouville equations

In his 1872 paper L. Boltzmann proved H-theorem for a class of kinetic equations. We generalize those equations in different manners in particular for quantum case. In 1906 H. Poincaré and in 2006 V. Kozlov and D. Treschev proposed new form of H-theorem for Liouville equation. We generalize their results in different directions. In particular we propose a theorem: time average coincides with Boltzmann extreme point. This theorem justifies (and corrects) the Gibbs method even in nonergodic case.