自然界复杂性的研究方法：熵的非唯一性
Approach of complexity in nature: Entropic nonuniqueness
摘要：
玻尔兹曼在19世纪70年代引入了一个对数度量，用来衡量系统的微观结构的熵和概率之间的联系。他的熵功能的经典系统是由吉布斯扩展到整个相空间的多体系统，并由冯诺依曼，以涵盖量子系统以及。最后，香农在信息论中使用了它。该泛函的最简单表达式对应于一组离散的$W$微观可能性，由$S_{BG}= -k\sum_{i=1}^W p_i \ln p_i$给出（$k$是一个正的普适常数; { \it BG}代表{ \it Boltzmann-Gibbs}）。这种关系使得BG统计力学的构造成为可能。BG理论提供了无数重要的应用。只要满足{ \it遍历性}的假设，它在物理系统中的应用是合理的.然而，{ \it当遍历性和类似的简单假设被违反时，我们能做什么？}，这确实发生在许多自然的、人工的和社会的复杂系统中。1988年提出了通过一族非加性熵（即$S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$）来推广BG统计力学的可能性，它在$q \to1$极限中恢复了加性$S_{BG}$熵。指数$q$由机械第一原理确定。三十年来，这一思想在世界范围内得到了深入的发展（参见\url{http://tsallis.cat.cbpf.br/biblio.htm}中的参考书目），并导致了大量的预测、验证和在物理系统和其他地方的应用。正如预期的那样，每当探索{ \it范式转换}时，社区中自然也会出现一些争议。本文从动力学和物理学基础出发，以最近的物理应用为结尾，介绍了这幅图景的现状。
英文摘要：
Boltzmann introduced in the 1870's a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His entropic functional for classical systems was extended by Gibbs to the entire phase space of a many-body system, and by von Neumann in order to cover quantum systems as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of $W$ microscopic possibilities, and is given by $S_{BG}= -k\sum_{i=1}^W p_i \ln p_i$ ($k$ is a positive universal constant; {\it BG} stands for {\it Boltzmann-Gibbs}). This relation enables the construction of BG statistical mechanics. The BG theory has provided uncountable important applications. Its application in physical systems is legitimate whenever the hypothesis of {\it ergodicity} is satisfied. However, {\it what can we do when ergodicity and similar simple hypotheses are violated?}, which indeed happens in very many natural, artificial and social complex systems. It was advanced in 1988 the possibility of generalizing BG statistical mechanics through a family of nonadditive entropies, namely $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$, which recovers the additive $S_{BG}$ entropy in the $q \to1$ limit. The index $q$ is to be determined from mechanical first principles. Along three decades, this idea intensively evolved world-wide (see Bibliography in \url{http://tsallis.cat.cbpf.br/biblio.htm}), and led to a plethora of predictions, verifications, and applications in physical systems and elsewhere. As expected whenever a {\it paradigm shift} is explored, some controversy naturally emerged as well in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations, and ending with its most recent physical applications.
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