Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.,Physical Review E

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Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.
Physical Review E ( IF 2.2 ) Pub Date : 2020-07-08 , DOI: 10.1103/physreve.102.012118
Jesús Fuentes ₁ , José Luis López ₁ , Octavio Obregón ₁

Affiliation

We derive generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures

S_{\pm}

that depend exclusively on the probability. These entropies have been originally obtained from the superstatistics framework, therefore they regard nonequilibrium systems outlined by a long-term stationary state in view of a spatiotemporally fluctuating intensive quantity. Moreover, entropies

S_{\pm}

as well as Boltzmann-Gibbs (BG) entropy

S_{B}

both pertain to the same asymptotical equivalence class, thus suggesting that

S_{\pm}

could depict a consistent thermodynamic generalization of BG. For these reasons, we assert that transport phenomena to be accounted for by our models shall coincide with the portrait given by the conventional FPEs for systems comprehending short-range interactions or a high number of accessible microstates, whereas, for systems composed of a small number of microstates, or those with long-range interactions, the governing equations of motion are to be the FPEs here derived, as long as the system fulfills the attributes mentioned above. We discuss the anomalous diffusion exhibited by the two generalized FPEs and also present some numerical applications. In particular, we find that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of our models.

中文翻译：

从非扩展熵渐近等效于Boltzmann-Gibbs的广义Fokker-Planck方程。

我们基于两个非广义熵测度推导广义的Fokker-Planck方程（FPE）

{小号}_{\pm}

这完全取决于概率。这些熵最初是从超统计框架中获得的，因此考虑到时空波动的集约量，它们会考虑由长期静止状态概述的非平衡系统。而且，熵

{小号}_{\pm}

以及玻尔兹曼-吉布斯（BG）熵

{小号}_{乙}

两者都属于同一渐近对等类，因此表明

{小号}_{\pm}

可以描述BG的一致热力学概括。出于这些原因，我们断言，对于包含短距离相互作用或大量可访问微状态的系统，我们的模型要考虑的传输现象应与常规FPE给出的肖像重合，而对于由少量FPE组成的系统如果微状态或具有长距离相互作用的状态，则运动的控制方程式就是此处导出的FPE，只要系统满足上述属性即可。我们讨论了由两个广义FPE表现出的反常扩散，并提出了一些数值应用。特别是，我们发现有一些有关生物学的模型，用于研究聚集和聚集行为，其结构与我们的模型之一相吻合。

更新日期：2020-07-08

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