Statistical mechanics of generalized conservative systems: self-organization by non-integrable topological constraints and non-elliptic diffusion processes

广义保守系统的统计力学：不可积拓扑约束和非椭圆扩散过程的自组织

基本信息

批准号：
18J01729
负责人：
佐藤直木
金额：
$ 2.33万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2018
资助国家：
日本
起止时间：
2018-04-25 至 2021-03-31
项目状态：
已结题

项目摘要

The aim of the present research is to formulate the statistical theory of mechanical systems subject to non-integrable topological constraints, and to create the mathematical objects, concepts, and methods that are required to achieve this goal. The following results were obtained:1、Ideal systems exhibit a Poisson structure. However, the algebraic structure of non-ideal systems is an open issue. Here, we showed that the Fokker-Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth.2、The statistical properties of topologically constrained mechanical systems can be related to the geometric properties of stationary solutions of the ideal Euler equations. Here, we investigated the existence of stationary solutions of the Euler equations without continuous Euclidean symmetries and with non-vanishing pressure gradients, and provided smooth analytic examples in bounded domains.3、The Sobolev-like Hilbert space of the solutions of the second order degenerate-elliptic partial differential equation (orthogonal Poisson equation) object of this study and the associated topology were examined. These results were reported in N Sato and Z Yoshida 2019 J. Phys. A: Math. Theor. 52 355202. Here, it is found that a non-vanishing helicity compensates the broken ellipticity.4、The existence of Beltrami operators in dimensions higher than 3 was shown. Analytic examples were given.

本研究的目的是建立受不可积拓扑约束的机械系统的统计理论，并创建实现这一目标所需的数学对象、概念和方法，获得了以下结果：1、理想系统呈现出泊松结构，然而，非理想系统的代数结构是一个悬而未决的问题，在这里，我们证明了描述非正则哈密顿系统中的扩散过程的福克-普朗克方程呈现出三重性。结构，即生成与能量守恒和熵增长的热力学原理一致的方程的代数形式。 2、拓扑约束机械系统的统计特性可以与理想欧拉方程的平稳解的几何特性相关。，我们研究了不具有连续欧几里德对称性且压力梯度不为零的欧拉方程平稳解的存在性，并提供了有界域中的平滑解析示例。3、本研究的二阶简并椭圆偏微分方程（正交泊松方程）的解的类 Sobolev 希尔伯特空间和相关拓扑已在 N Sato 和 Z Yoshida 2019 J. Phys A 中报告。：数学。 52 355202。这里，发现不消失的螺旋性补偿了破碎的椭圆性。4，存在性给出了维度大于 3 的 Beltrami 算子的解析示例。