Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations

Abstract

A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard -- linear Fokker-Planck equation -- may be also related to a family of nonlinear Fokker-Planck equations.

19 pages, no figures

References in corpus (1)

• A General Nonlinear Fokker-Planck Equation and its Associated Entropy

Cited by in corpus (17)

• Thermostatistics of overdamped motion of interacting particles
• Analytic approaches of the anomalous diffusion: a review
• q-Gaussians in the porous-medium equation: stability and time evolution
• Curl Forces and the Nonlinear Fokker-Planck Equation
• Thermodynamic Framework for Compact q-Gaussian Distributions
• Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
• Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein
• Microscopic Dynamics of Nonlinear Fokker-Planck Equations
• Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution
• Stationary solution and $H$ theorem for a generalized Fokker-Planck equation
• Ehrenfest urn model with interaction
• Phase Transitions in Ehrenfest Urns Model with Interactions: Coexistence of uniform and non-uniform states
• Singular diffusion and criticality in a confined sandpile
• Non-equilibrium thermodynamics and Phase transition of Ehrenfest urns with interactions
• A unified framework for divergences, free energies, and Fokker-Planck equations
• Limit distribution in the $q$-CLT for $q\ge 1$ can not have a compact support
• A Nonlinear $q$-Deformed Schrödinger Equation