Tsallis entropy and nonextensive generalization of Boltzmann-Gibbs statistical mechanics

在2000年和2001年，我一直从热力学和信息论的角度对上述研究项目进行研究。我推导出了最一般的熵的组成定律，它与“可组合性”的概念和热力学的第零定律是相容的。从由此得到的非加性熵开始，我建立了热力学的勒让德变换结构。这使我们能够计算非扩展系统的热力学性质。我还研究了宏观热力学平衡的一般性质，发现这种状态不仅可以用标准玻尔兹曼因子来表征，而且可以用tsallis型幂律因子来表征。然后，发展了非加性量子信息理论。我发现这个新理论在某些方面优于普通的加性冯·诺依曼理论。最后，我讨论了与幂律波包中位置和动量观测值的测量相关的信息熵不确定性关系，这可能有发散的方差。我可以证明，这种信息理论方法可以揭示波包的性质，这是不能用标准的海森堡类型的不确定性关系公式来讨论的。

In 2000 and 2001, I have been addressing to the above research project from the viewpoints of thermodynamics and information theory. I have derived the most general composition law of entropy, which is compatible with the concept of "composability" and the zeroth law of thermodynamics. Starting with thus obtained nonadditive entropy, I have established the thermodynamic Legendre transform structure. This enables one to calculate thermodynamic properties of nonextensive systems. I have also studied the general properties of macroscopic thermodynamical equilibrium, and have found that such a state can be characterized not only by the standard Boltzmann factor but also by the Tsallis-type power-law factor. Then, nonadditive quantum information theory has been developed. I have found that this new theory has in some points superior to the ordinary additive von Neumann theory. Finally, I have discussed the information entropic uncertainty relation associated with the measurements of the position and momentum observables in the power-law wave packets, which may have the divergent variances. I could show that this information theoretic approach can reveal the properties of the wave packets, which cannot be discussed by using the standard Heisenberg-type formulation of the uncertainty relation.