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.

中文翻译：

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广义福克-普朗克方程的平稳解和定理

我们研究了一系列广义 Fokker-Planck 方程，其中包含理查森方程和多孔介质方程作为成员。考虑与有效势相关的限制漂移项，我们表明该族的每个方程都有一个取决于该势的平稳解。这个平稳解包含几个众所周知的概率分布。此外，我们验证了一个$H$使用类自由能泛函的广义​​ Fokker-Planck 方程的定理。我们表明，每个泛函的类能量部分基于有效势，类熵部分是广义的 Tsallis 熵形式，它对位置具有不寻常的依赖性，并且可以与 Kullback-Leibler 的泛化有关分歧。我们还验证了这种受方便约束的熵状形式的优化可以恢复平稳解。这里提出的分析包括几项关于$H$ 其他广义 Fokker-Planck 方程的定理作为特殊情况。