Jauregui Max, Lucchi Anna L F, Passos Jean H Y, Mendes Renio S
Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo, 5790 CEP 87020-900 - Maringá - PR - Brazil.
Departamento de Física, Universidade Estadual de Maringá, Av. Colombo, 5790 CEP 87020-900 - Maringá - PR - Brazil.
Phys Rev E. 2021 Sep;104(3-1):034130. doi: 10.1103/PhysRevE.104.034130.
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.
我们研究了一族广义福克-普朗克方程，其中包括理查森方程和多孔介质方程。考虑与有效势相关的限制漂移项，我们证明该族中的每个方程都有一个依赖于该势的稳态解。这个稳态解包含了几个著名的概率分布。此外，我们使用类自由能泛函验证了广义福克-普朗克方程的一个H定理。我们表明，每个泛函的能量类部分基于有效势，而熵类部分是一种广义的Tsallis熵形式，它对位置有不寻常的依赖，并且可以与库尔贝克-莱布勒散度的推广相关。我们还验证了在适当约束下对这种类熵形式的优化可以恢复稳态解。这里给出的分析包括将关于其他广义福克-普朗克方程的几个H定理研究作为特殊情况。
