arXiv 1607.070972016-07-24cond-mat.stat-mechmath-phmath.MP

复杂系统的统计力学：关于$q$ -三元组的结构

Statistical mechanics for complex systems: On the structure of $q$-triplets

Constantino Tsallis

备注：: Invited contribution to the Proceedings of the 31st International Colloquium on Group Theoretical Methods in Physics (Rio de Janeiro, 2016); 7 pages

摘要：

大量的自然、人工和社会复杂系统的存在违反了基本假设（例如，Boltzmann-Gibbs（BG）统计力学的遍历性。通过引入非加性熵泛函，如$S_q\equiv k\frac{1-\sum_{i=1}^W p_i^q}{q-1} \; \Bigl(q \in {\cal R}; \, \sum_{i=1}^W p_i=1 \Bigr)$和$S_1=S_{BG}\equiv -k\sum_{i=1}^W p_i \ln p_i$，可以令人满意地处理许多这样的情况。每一类这样的系统可以由一组值$\{q\}$来表征，这些值直接对应于其各种物理/动力学/几何性质。一个最重要的子集通常被称为$q$三元组，即$(q_{sensitivity}, q_{relaxation}, q_{stationary\,state})$，在本文的主体中定义。在BG限值中，我们有$q_{sensitivity}=q_{relaxation}=q_{stationary\,state}=1$。对于给定的一类复杂系统，集合$\{q\}$仅包含$q$的几个独立值，所有其他值都是这几个值的函数。该结构的说明在2005年给出[Tsallis，Gell-Mann和Sato，Proc.Natl. Acad. Sc. USA { \bf 102}，15377; TGS]。这幅插图使旅行者1号关于太阳风的数据得到了令人满意的分析。但这些结构的一般形式仍然是一个悬而未决的问题。例如，对于与逻辑斯蒂映射的混沌边缘相关联的具有挑战性的$q$ -三元组，情况就是如此。我们在这里介绍一个变换，它合理地推广了TGS的一个，这可能是迈向一般解决方案的重要一步。

英文摘要：

A plethora of natural, artificial and social complex systems exists which violate the basic hypothesis (e.g., ergodicity) of Boltzmann-Gibbs (BG) statistical mechanics. Many of such cases can be satisfactorily handled by introducing nonadditive entropic functionals, such as $S_q\equiv k\frac{1-\sum_{i=1}^W p_i^q}{q-1} \; \Bigl(q \in {\cal R}; \, \sum_{i=1}^W p_i=1 \Bigr)$, with $S_1=S_{BG}\equiv -k\sum_{i=1}^W p_i \ln p_i$. Each class of such systems can be characterized by a set of values $\{q\}$, directly corresponding to its various physical/dynamical/geometrical properties. A most important subset is usually referred to as the $q$-triplet, namely $(q_{sensitivity}, q_{relaxation}, q_{stationary\,state})$, defined in the body of this paper. In the BG limit we have $q_{sensitivity}=q_{relaxation}=q_{stationary\,state}=1$. For a given class of complex systems, the set $\{q\}$ contains only a few independent values of $q$, all the others being functions of those few. An illustration of this structure was given in 2005 [Tsallis, Gell-Mann and Sato, Proc. Natl. Acad. Sc. USA {\bf 102}, 15377; TGS]. This illustration enabled a satisfactory analysis of the Voyager 1 data on the solar wind. But the general form of these structures still is an open question. This is so, for instance, for the challenging $q$-triplet associated with the edge of chaos of the logistic map. We introduce here a transformation which sensibly generalizes the TGS one, and which might constitute an important step towards the general solution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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