1212.6767 (Alexander N. Gorban)

General H-theorem and entropies that violate the second law    [PDF]

Alexander N. Gorban

We study systems with finite number of states $A_i$, which obey the first order kinetics (master equation). A general criterion is found for the existence of $H$-theorem with given $H$. A convex function $H$ is a Lyapunov function for all master equations with the given equilibrium if and only if its conditional minima properly describe the equilibira of pair transitions $A_i \rightleftharpoons A_j$. This theorem does nod depend on the principle of detailed balance and is valid both for reversible and for general Markov kinetics. Analysis of pair equilibria demonstrates, for example, that the popular Bregman divergences like Euclidian distance or Itakura--Saito divergence in the space of distributions cannot be the universal Lyapunov functions for the first-order kinetics and increase in some Markov processes.

View original: http://arxiv.org/abs/1212.6767