广义熵是什么样的？复杂非遍历系统的公理化方法
What do generalized entropies look like? An axiomatic approach for complex, non-ergodic systems
摘要：
Shannon和Khinchin指出，假设有四个信息论公理，熵必须是Boltzmann-Gibbs型的，即$S=-\sum_i p_i \log p_i$。这里我们注意到，在物理系统中，这些公理之一可能被违反。对于非遍历系统，所谓的分离公理（Shannon-Khinchin公理4）通常是无效的。我们发现，当这个公理被违反的熵采取更一般的形式，$S_{c,d}\propto \sum_i ^W Γ(d+1, 1- c \log p_i)$，其中$c$和$d$是标度指数和$Γ(a,b)$是不完全的伽玛函数。指数$(c,d)$定义了所有相互作用和非相互作用系统的等价类，并明确地描述了任何统计系统的热力学极限。证明是可能的，因为两个新发现的标度定律，任何熵形式必须满足，如果前三个香农-欣钦公理成立。$(c,d)$可用于定义统计系统的等价类。一系列已知的熵可以根据这些等价类进行分类。我们表明，相应的分布函数是特殊形式的朗伯- ${\cal W}$指数包含-作为特殊情况-玻尔兹曼，拉伸指数和Tsallis分布（幂律）。在推导中，我们假设迹形式的熵，$S=\sum_i g(p_i)$，与$g$一些功能，然而，更一般的熵形式可以沿着相同的标度分析进行分类。
英文摘要：
Shannon and Khinchin showed that assuming four information theoretic axioms the entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here we note that in physical systems one of these axioms may be violated. For non-ergodic systems the so called separation axiom (Shannon-Khinchin axiom 4) will in general not be valid. We show that when this axiom is violated the entropy takes a more general form, $S_{c,d}\propto \sum_i ^W Γ(d+1, 1- c \log p_i)$, where $c$ and $d$ are scaling exponents and $Γ(a,b)$ is the incomplete gamma function. The exponents $(c,d)$ define equivalence classes for all interacting and non interacting systems and unambiguously characterize any statistical system in its thermodynamic limit. The proof is possible because of two newly discovered scaling laws which any entropic form has to fulfill, if the first three Shannon-Khinchin axioms hold. $(c,d)$ can be used to define equivalence classes of statistical systems. A series of known entropies can be classified in terms of these equivalence classes. We show that the corresponding distribution functions are special forms of Lambert-${\cal W}$ exponentials containing -- as special cases -- Boltzmann, stretched exponential and Tsallis distributions (power-laws). In the derivation we assume trace form entropies, $S=\sum_i g(p_i)$, with $g$ some function, however more general entropic forms can be classified along the same scaling analysis.
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