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

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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.