Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot

TL;DR

This paper introduces a new class of generalized Fokker-Planck equations that conserve key physical quantities and maximize a generalized entropy, with applications spanning astrophysics, fluid dynamics, and biological systems, offering a systematic framework for modeling complex relaxation processes.

Contribution

The paper develops a novel relaxation equation based on generalized thermodynamics, unifying various entropy forms, and applies it to diverse physical systems including stellar and vortex dynamics.

Findings

New relaxation equation depends on a key skewness parameter.

Standard parametrizations are special cases at infinite temperature.

The approach links vortices, stars, and bacteria through entropy classification.

Abstract

We introduce a new set of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy until a maximum entropy state is reached. The concept of generalized entropies is rigorously justified for continuous Hamiltonian systems undergoing violent relaxation. Tsallis entropies are just a special case of this generalized thermodynamics. Application of these results to stellar dynamics, vortex dynamics and Jupiter's great red spot are proposed. Our prime result is a novel relaxation equation that should offer an easily implementable parametrization of geophysical turbulence. This relaxation equation depends on a single key parameter related to the skewness of the fine-grained vorticity distribution. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations may have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in classes of equivalence and provide an aesthetic connexion between topics (vortices, stars, bacteries,...) which were previously disconnected.