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，我一直从热力学和信息理论的角度来解决上述研究项目。我得出了最一般的熵成分定律，该定律与“合成性”和热力学的零度概念兼容。从因此获得的非熵熵开始，我已经建立了热力学Legendre变换结构。这使一个人能够计算非xtentandistion系统的热力学特性。我还研究了宏观热力学平衡的一般特性，并发现这种状态不仅可以由标准的玻尔兹曼因子来表征，而且可以由tsallis型幂律因子来表征。然后，已经开发了非添加量子信息理论。我发现，在某些角度，这种新理论优于普通的添加剂von Neumann理论。最后，我讨论了与幂律波浪数据包中位置和动量可观察物的测量相关的信息熵不确定性关系，这些位置和动量可观察到可能具有不同的方差。我可以表明，这种信息理论方法可以揭示波数据包的特性，这无法通过使用不确定性关系的标准海森堡型公式来讨论。

项目成果

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 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: "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)