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arXiv · 2109.06237

Stationary solution and $H$ theorem for a generalized Fokker-Planck equation

Max Jauregui·Anna L. F. Lucchi·Jean H. Y. Passos·Renio S. Mendes
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Abstract

We investigate a family of generalized Fokker-Planck equations that contains Richardson and porous media equations as members. Considering a confining drift term that is related to an effective potential, we show that each equation of this family has a stationary solution that depends on this potential. This stationary solution encompasses several well-known probability distributions. Moreover, we verify an $H$ theorem for the generalized Fokker-Planck equations using free-energy-like functionals. We show that the energy-like part of each functional is based on the effective potential and the entropy-like part is a generalized Tsallis entropic form, which has an unusual dependence on the position and can be related to a generalization of the Kullback-Leibler divergence. We also verify that the optimization of this entropic-like form subjected to convenient constraints recovers the stationary solution. The analysis presented here includes several studies about $H$ theorems for other generalized Fokker-Planck equations as particular cases.

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Max Jauregui, Anna L. F. Lucchi, Jean H. Y. Passos, Renio S. Mendes. 2021-09-13. Stationary solution and $H$ theorem for a generalized Fokker-Planck equation. https://doi.org/10.1103/physreve.104.034130

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