Tsallis entropy and nonextensive generalization of Boltzmann-Gibbs statistical mechanics

Tsallis 熵和 Boltzmann-Gibbs 统计力学的非广延推广

• 批准号: 12640381
• 负责人: ABE Sumiyoshi
• 金额: $ 0.77万
• 依托单位: University of Tsukuba (2001)Nihon University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640381/
• 关键词: Tsallis statistics; nonadditive entropy; thermodynamic formalism; quantum entanglement; information theoretic uncertainty; 熱力学形式; 量子エンタングルメント; Tsallisのnonextensive統計力学; 非加法的量子情報理論; 一般化された熱力学

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.

在2000年和2001年，我一直致力于从热力学和信息论的观点对上述研究项目。我导出了最一般的熵的合成定律，它与“可合成性”的概念和热力学第零定律相容。从这样得到的非加性熵开始，我建立了热力学勒让德变换结构。这使得人们能够计算非广延系统的热力学性质。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比普通的加性冯·诺依曼理论有上级之处。最后，讨论了幂律波包中位置和动量观测量的信息熵测不准关系，这些观测量可能具有发散方差。我可以证明，这种信息论方法可以揭示波包的性质，这不能通过使用标准的海森堡型公式的不确定关系进行讨论。

项目成果

S.Abe and A.K.Rajagopal: "Microcanonical foundation for systems with power-law distributions"Journal of Physics A. 33. 8733-8738 (2000)

S.Abe 和 A.K.Rajagopal：“幂律分布系统的微正则基础”物理学杂志 A. 33. 8733-8738 (2000)

S.Abe and Y.Okamoto 共編著: "Nonextensive Statistical Mechanics and Its Applications"Springer-Verlag. 279 (2001)

S.Abe 和 Y.Okamoto 共同编辑：“非广延统计力学及其应用”Springer-Verlag 279 (2001)。

S.Abe: "General pseudoadditivity of composable entropy prescribed by the existence of eguilibrium"Physical Review E. 63. 061105 (2001)

S.Abe：“由平衡的存在规定的可组合熵的一般伪可加性”物理评论 E. 63. 061105 (2001)

S.Abe, Y.Okamoto共編著: "Nonextensive Statistical Mechanics and Its Applications"Springer-Verlag. 277 (2001)

S.Abe 和 Y.Okamoto 共同编辑：“非广延统计力学及其应用”Springer-Verlag 277 (2001)。

S.Abe and A.K.Rajagopal: "Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half-spaces"Journal of Physics A. 33. 8723-8732 (2000)

S.Abe 和 A.K.Rajagopal：“全空间和半空间中非广延统计分布与 Levy 分布的收敛率”物理学杂志 A. 33. 8723-8732 (2000)