标题： 组熵、关联规律和ζ函数 显示英文标题

标题： Group entropies, correlation laws and zeta functions

摘要： 提出了群熵的概念。 它能够统一和推广文献中已知的许多不同定义的熵，例如玻尔兹曼-吉布斯、塔拉萨斯、阿贝和卡尼阿迪斯的熵。 还介绍了其他新的熵泛函，这些泛函与描述非平衡系统普适类的非平凡关联规律有关，当动力学表现为弱混沌时尤为如此。 讨论了相关的热统计学。 我们构造的数学基础是形式群论，它提供了粒子之间关联的一般结构，并决定了相应的熵泛函。 作为一个应用示例，展示了群熵在信息论中的作用，并提出了Kullback-Leibler散度的推广形式。 建立了统计力学与泽塔函数之间的一种新联系。 特别是，塔拉萨斯熵与经典的黎曼泽塔函数相关联。 显示英文摘要

摘要： The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.