非广延熵的仿射扩张与统计力学
Affinity-based extension of non-extensive entropy and statistical mechanics
摘要：
Tsallis的非广延熵扩展到包含系统的微观状态之间的亲和力的依赖。在我们构建扩展熵（$\mathcal{H}$）的核心是不同状态的有效数量的概念，称为有效多样性（$\mathitΔ$）。它是由状态间的概率分布和状态间的仿射关系导出的唯一的综合测度。有效分集通过玻尔兹曼方程式关系$\mathcal{H}=\ln_{q}\mathitΔ$与扩展熵相关，以Tsallis的$q$ -对数表示。一个新的原则称为嵌套原则的建立，指出有效的多样性保持不变的任意分组的组成状态。结果表明，这种不变性仅适用于$q=2$，然而，不变性恢复一般$q$在零亲和极限（即Tsallis和Boltzmann-Gibbs的情况下）。利用基于仿射的扩展Tsallis熵，在一般态间仿射存在的情况下，构造了微正则系综和正则系综。结果表明，经典的假设，先验概率相等不再成立，但修改的仿射依赖项。作为例子，我们用扩展正则方法研究了一个二能级系统，结果表明，态间亲和势对平衡态热力学量的热行为有影响。此外，附录中还讨论了基于亲和力的扩展多样性/熵在信息论和生物多样性理论中的一些应用和含义。
英文摘要：
Tsallis' non-extensive entropy is extended to incorporate the dependence on affinities between the microstates of a system. At the core of our construction of the extended entropy ($\mathcal{H}$) is the concept of the effective number of dissimilar states, termed the effective diversity ($\mathitΔ$). It is a unique integrated measure derived from the probability distribution among states and the affinities between states. The effective diversity is related to the extended entropy through the Boltzmann's-equation-like relation, $\mathcal{H}=\ln_{q}\mathitΔ$, in terms of the Tsallis' $q$-logarithm. A new principle called the Nesting Principle is established, stating that the effective diversity remains invariant under an arbitrary grouping of the constituent states. It is shown that this invariance property holds only for $q=2$; however, the invariance is recovered for general $q$ in the zero-affinity limit (i.e. the Tsallis and Boltzmann-Gibbs case). Using the affinity-based extended Tsallis entropy, the microcanonical and the canonical ensembles are constructed in the presence of general between-state affinities. It is shown that the classic postulate of equal a priori probabilities no longer holds but is modified by affinity-dependent terms. As an illustration, a two-level system is investigated by the extended canonical method, which manifests that the thermal behaviours of the thermodynamic quantities at equilibrium are affected by the between-state affinity. Furthermore, some applications and implications of the affinity-based extended diversity/entropy for information theory and biodiversity theory are addressed in appendices.
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