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

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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.