The concept of duality of probability distributions constitutes a fundamental “brick” in the solid framework of nonextensive statistical mechanics—the generalization of Boltzmann–Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb’s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.

中文翻译：

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非广延统计力学：双熵与双概率之间的等价


概率分布对偶​​性的概念构成了非广延统计力学坚实框架中的一个基本“砖块”——考虑q熵的玻尔兹曼-吉布斯统计力学的推广。概率对偶性正在解决该理论中长期存在的问题，例如，在给定微观态能量的可加性的情况下，它确定内能的可加性。然而，它是理论中相当复杂的一部分，当然，它不能沿着熵最大化的吉布路径简单地解释。最近，研究表明存在另一种情况，考虑对偶熵而不是对偶概率。特别是，非广延统计力学的框架可以使用 q- 和 1/q- 熵等效地开发。规范概率分布再次与已知的 q 指数分布一致，但不需要普通护航概率的对偶性。此外，结果表明，对偶熵、q 熵和 1/q 熵以及 1 熵都涉及一个恒等式，这在理论发展和应用中非常有用。