Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence

2002/09/30 by Pierre-Henri Chavanis · 11 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Fokker–Planck equation #Fractional Differential Equations Solutions #Partial differential equation #Physics #Planck #Quantum mechanics #Statistical Mechanics and Entropy #Statistical physics #Thermodynamics #Tsallis entropy #Tsallis statistics #Turbulence #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.68.036108

published as Phys. Rev. E, 68, 036108 (2003) · Submitted to Phys. Rev. E

arxiv created 2003/05/02 · openalex publication_date 2003/09/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. 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 can 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 connection between topics (vortices, stars, bacteria, em leader ) which were previously disconnected.

Citations

• Statistical mechanics of the shallow water system
• Thermodynamics and collapse of self-gravitating Brownian particles inDdimensions

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