da Costa Bruno G, Gomez Ignacio S, Borges Ernesto P
Instituto Federal de Educação, Ciência e Tecnologia do Sertão Pernambucano, Rua Maria Luiza de Araújo Gomes Cabral s/n, 56316-686 Petrolina, Pernambuco, Brazil.
Instituto de Fisica, Universidade Federal da Bahia, R. Barao de Jeremoabo s/n, 40170-115 Salvador, Bahia, Brazil.
Phys Rev E. 2020 Dec;102(6-1):062105. doi: 10.1103/PhysRevE.102.062105.
We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.
我们通过利用朗之万方程，在任意变形空间的广义变形导数的背景下，给出了具有位置依赖质量粒子的非均匀介质的福克 - 普朗克方程（FPE），其中FPE的线性（非线性）特征与所采用的变形线性（非线性）导数相关。具有位置依赖扩散系数的非均匀介质的FPE等同于变形空间内的变形FPE，该变形空间由广义导数和恒定扩散系数描述。当温度和迁移率具有相同的位置依赖函数形式时，变形FPE与非均匀介质的扩散方程一致，并且与非线性朗之万方法也一致。H定理的变形形式允许将玻尔兹曼 - 吉布斯熵泛函表示为两个贡献之和，一个来自粒子，另一个来自非均匀介质。用无限深方势阱和具有线性漂移系数的限制势对该形式体系进行了说明。还讨论了超统计与位置依赖朗之万方程之间的联系。
