General H-theorem and Entropies that Violate the Second Law
General H-theorem and Entropies that Violate the Second Law
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DOI：
10.3390/e16052408
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发表时间：
2014-05-01
期刊：
影响因子：
2.7
通讯作者：
Gorban, Alexander N.
中科院分区：
文献类型：
--
作者：
Gorban, Alexander N.
H定理指出熵产生是非负的，因此，封闭系统的熵应该随时间单调变化。在信息处理中，对于信号的随机变换，熵产生是正的（信息处理引理）。最初，经典的Boltzmann-Gibbs-Shannon熵和相应的散度（相对熵）证明了H定理和信息处理引理。在过去的几十年里，人们提出了许多新的熵和发散，对于所有这些熵和发散，都需要H-定理。本文提出了一个简单而普遍的判别准则来检验H定理对凸散度H是否成立，并证明了一些流行的散度不服从H定理。我们考虑服从一阶动力学（主方程）的n个状态A（i）的系统。凸函数H是所有具有给定平衡点的主方程的李雅普诺夫函数当且仅当其条件极小值恰当地描述了对转移A（i）（sic）A（j）的平衡点。这个定理不依赖于细节平衡原理，对一般的马尔可夫动力学是有效的。对平衡点的初步分析表明，流行的Bregman分歧，如欧几里德距离或Itakura-Saito距离在分布空间不能是通用的李雅普诺夫函数的一级动力学，可以增加在马尔可夫过程。因此，它们违反了第二定律和信息处理引理。特别是，对于这些信息（分歧）的措施随机操纵数据可能会增加信息的数据。主要结果推广到非线性广义质量作用定律动力学方程。
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) (sic) 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 demonstrate that the popular Bregman divergences like Euclidian 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.