复杂统计系统的综合分类及其熵和分布函数的从头算推导
A comprehensive classification of complex statistical systems and an ab-initio derivation of their entropy and distribution functions
摘要：
为了在数学框架内描述强相互作用的统计系统--特别是复杂系统--可能有必要引入广义熵$S_g$。过去已经提出了一系列这样的熵，主要是为了适应重要的经验分布函数的最大无知原则。到目前为止，对这些熵的基本起源及其与复杂系统的更深层次关系的理解是有限的。在这里，我们从第一原理探讨这个问题。我们首先观察到一般强相互作用系统违反了第四个Khinchin公理（可分性公理），并询问在假设前三个Khinchin公理（K1-K3）成立和$S_g=\sum_ig(p_i)$的情况下违反第四个公理的后果。我们证明了简单的缩放参数，在这些要求下，{ \em每个}统计系统是唯一的特点是一个独特的一对标度指数$(c,d)$在大尺寸的限制。指数定义所有相互作用和非相互作用系统的等价类。这允许导出唯一的熵$S_{c,d}\propto \sum_i Γ(d+1, 1- c \ln p_i)$，其覆盖关于K1-K3的所有熵并且可以被写为$S_g=\sum_ig(p_i)$。已知的熵现在可以在这些等价类中分类。相应的分布函数是朗伯- $W$指数的特殊形式，包含玻尔兹曼分布、拉伸指数分布和Tsallis分布（幂律）的特殊情况--所有这些分布在自然界中都非常丰富。据我们所知，这是广义熵存在的第一个{ \em ab initio}证明。即使在这里，我们假设$S_g=\sum_ig(p_i)$，我们表明，更一般的熵的形式可以沿着相同的路线进行分类。
英文摘要：
To characterize strongly interacting statistical systems within a thermodynamical framework - complex systems in particular - it might be necessary to introduce generalized entropies, $S_g$. A series of such entropies have been proposed in the past, mainly to accommodate important empirical distribution functions to a maximum ignorance principle. Until now the understanding of the fundamental origin of these entropies and its deeper relations to complex systems is limited. Here we explore this questions from first principles. We start by observing that the 4th Khinchin axiom (separability axiom) is violated by strongly interacting systems in general and ask about the consequences of violating the 4th axiom while assuming the first three Khinchin axioms (K1-K3) to hold and $S_g=\sum_ig(p_i)$. We prove by simple scaling arguments that under these requirements {\em each} statistical system is uniquely characterized by a distinct pair of scaling exponents $(c,d)$ in the large size limit. The exponents define equivalence classes for all interacting and non interacting systems. This allows to derive a unique entropy, $S_{c,d}\propto \sum_i Γ(d+1, 1- c \ln p_i)$, which covers all entropies which respect K1-K3 and can be written as $S_g=\sum_ig(p_i)$. Known entropies can now be classified within these equivalence classes. The corresponding distribution functions are special forms of Lambert-$W$ exponentials containing as special cases Boltzmann, stretched exponential and Tsallis distributions (power-laws) -- all widely abundant in nature. This is, to our knowledge, the first {\em ab initio} justification for the existence of generalized entropies. Even though here we assume $S_g=\sum_ig(p_i)$, we show that more general entropic forms can be classified along the same lines.
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