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

本研究的目的是制定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。得到以下结果：1、理想体系具有泊松结构。然而，非理想系统的代数结构是一个开放的问题。在这里，我们证明了描述非正则哈密顿系统中扩散过程的Fokker-Planck方程呈现出一种三重结构，即一种代数形式，它产生的方程符合能量守恒和熵增长的热力学原理。拓扑约束机械系统的统计性质与理想欧拉方程的定态解的几何性质有关。本文研究了无连续欧几里德对称且压力梯度不消失的Euler方程的平稳解的存在性，并给出了有界域上的光滑解析实例。3、研究了本文研究对象二阶退化椭圆型偏微分方程（正交泊松方程）解的Sobolev-like Hilbert空间及其拓扑结构。这些结果发表在N Sato和Z Yoshida 2019 J. Phys上。答:数学。理论52 355202。在这里，我们发现一个不消失的螺旋补偿了破碎的椭圆。4、证明了大于3维Beltrami算子的存在性。给出了解析实例。

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.