General H-theorem and entropies that violate the second law

Abstract

$H$-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the $H$-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the relative entropy). Many new entropies and divergences have been proposed during last decades and for all of them the $H$-theorem is needed. This note proposes a simple and general criterion to check whether the $H$-theorem is valid for a convex divergence $H$ and demonstrates that some of the popular divergences obey no $H$-theorem. We consider systems with $n$ states $A_i$ that obey first order kinetics (master equation). A convex function $H$ is a Lyapunov function for all master equations with given equilibrium if and only if its conditional minima properly describe the equilibria of pair transitions $A_i\rightleftharpoons A_j$. This theorem does not depend on the principle of detailed balance and is valid for general Markov kinetics. Elementary analysis of pair equilibria demonstrates that the popular Bregman divergences like Euclidean distance or Itakura-Saito distance in the space of distribution cannot be the universal Lyapunov functions for the first-order kinetics and can increase in Markov processes. Therefore, they violate the second law and the information processing lemma. In particular, for these measures of information (divergences) random manipulation with data may add information to data. The main results are extended to nonlinear generalized mass action law kinetic equations. In Appendix, a new family of the universal Lyapunov functions for the generalized mass action law kinetics is described.

45 pages, a significantly extended postprint version with description of a new family of the universal Lyapunov functions for the generalized mass action law kinetics. A presentation "New universal Lyapunov functions for nonlinear kinetics" is added as an ancillary file

References in corpus (11)

• Method of invariant manifold for chemical kinetics
• A geometric approach to the Global Attractor Conjecture
• Finding complex balanced and detailed balanced realizations of chemical reaction networks
• Thermodynamics in the Limit of Irreversible Reactions
• Uniqueness of thermodynamic projector and kinetic basis of molecular individualism
• Maxallent: Maximizers of all Entropies and Uncertainty of Uncertainty
• Legendre Integrators, Post-Processing and Quasiequilibrium
• Monotonically equivalent entropies and solution of additivity equation
• Finding weakly reversible realizations of chemical reaction networks using optimization
• Local Equivalence of Reversible and General Markov Kinetics
• Kinetic Path Summation, Multi--Sheeted Extension of Master Equation, and Evaluation of Ergodicity Coefficient

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• Bounds on the rates of statistical divergences and mutual information via stochastic thermodynamics
• Universal Gorban's Entropies: Geometric Case Study