Non-Equilibrium Statistical Mechanics with Topological Constraints: Thermodynamics and Entropy Production of Self-Organized Turbulence

具有拓扑约束的非平衡统计力学：自组织湍流的热力学和熵产生

• 批准号: 16J01486
• 负责人: 佐藤 直木
• 金额: $ 1.09万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2016
• 资助国家: 日本
• 起止时间: 2016-04-22 至 2018-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16J01486/
• 关键词: Conservative Dynamics; Topological Constraints; Diffusion; Entropy Measure; Invariant Measure; Helicity; Non-Integrability; Non-Elliptic PDE; Self-Organization; Hamiltonian Mechanics; Almost Poisson Algebra; Jacobi Identity

1、The theory developed in the present study was summarized in the Phd thesis with title "Generalized conservative dynamics in topologically constrained phase space: macro-hierarchy, entropy production, and self-organization". In this work a statistical theory of conservative systems subject to topological constraints is formulated, and the relationship between constraints and entropy measure is clarified by derivation of the H-theorem.2、The theory was applied to study plasma equilibria resulting by diffusion in different magnetic geometries: a straight magnetic field and a dipole magnetic field. Particle simulations confirmed the theoretical prediction that diffusion in a dipole magnetic field generate density gradients.3、In the study of diffusion processes subject to non-integrable constraints, it was found that there exists an additional mechanism to generate organized structures that does not arise when the constraints are integrable. Such self-organization is driven by a distortion of space caused by a geometrical charge, the "field charge". This quantity measures the departure of the operator (the antisymmetric matrix that generates the dynamics together with the Hamiltonian) from a Beltrami field.4、It was found that the stationary form of the diffusion equation describing systems affected by topological constraints is a non-elliptic second order partial differential equation. Specific conditions for existence and uniqueness of solution were obtained by using the integrability properties of the constraints.

1、本研究中的理论在博士论文“拓扑约束相空间中的广义保守动力学：宏观层次、熵产生和自组织”中得到了总结。本文建立了拓扑约束下保守系统的统计理论，并通过H定理的推导阐明了拓扑约束与熵测度之间的关系。2、应用该理论研究了不同磁几何中由扩散引起的等离子体平衡：直磁场和偶极磁场。粒子模拟证实了偶极磁场中扩散产生密度梯度的理论预言。3、在研究非可积约束条件下的扩散过程时，发现存在一种额外的机制来产生有序结构，而当约束条件是可积的时，这种机制是不存在的。这种自组织是由几何电荷（“场电荷”）引起的空间扭曲驱动的。这个量度量了算子（与哈密顿量一起产生动力学的反对称矩阵）与Beltrami场的偏离。4、发现描述系统受拓扑约束影响的扩散方程的定态形式是一个非椭圆二阶偏微分方程。利用约束条件的可积性，得到了解的存在唯一性的具体条件。

项目成果

測度保存型括弧積を持つ準ハミルトン力学系の統計力学

具有测度守恒括号的准哈密尔顿动力系统的统计力学

• 发表时间: 2017
• 作者: 佐藤直木;吉田善章
• 通讯作者: 吉田善章

Relaxation and Stability of Compressible Euler Flow in a Toroidal Domain

环形域中可压缩欧拉流的弛豫与稳定性

• 发表时间: 2017
• 作者: Kenji Mishima;Hirochika Sumino;Takahito Yamada;Sei Ieki;Naoki Nagakura;Hidetoshi Otono;Hideyuki Oide;梶ヶ谷徹;R. L. Dewar and N. Sato;Naoki Nagakura Kazuki Fujii Isao Harayama Yu Kato Daiichiro Sekiba Yumi Watahiki Satoru Yamashita;國川 慶太;N. Sato and Z. Yoshida;國川 慶太;Naoki Nagakura;N. Sato and R. L. Dewar
• 通讯作者: N. Sato and R. L. Dewar

ヤコビ律を満たさない括弧積の拡張によるポアソン括弧

由于括号乘积展开不满足雅可比规则而导致泊松括号

• 发表时间: 2016
• 作者: 佐藤直木;吉田善章
• 通讯作者: 吉田善章

Theoretical Modeling and Particle Simulation of Inward Diffusion in a Radiation Belt

辐射带内向扩散的理论建模和粒子模拟

• 发表时间: 2018
• 作者: Y. Ushida;Y. Kawazura;N. Sato;and Z. Yoshida;國川慶太;N. Sato and Z. Yoshida
• 通讯作者: N. Sato and Z. Yoshida

Recirculating Flow of an Euler Fluid

欧拉流体的再循环流动

• 发表时间: 2017
• 作者: Kenji Mishima;Hirochika Sumino;Takahito Yamada;Sei Ieki;Naoki Nagakura;Hidetoshi Otono;Hideyuki Oide;梶ヶ谷徹;R. L. Dewar and N. Sato
• 通讯作者: R. L. Dewar and N. Sato