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.

中文翻译：

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变形的Fokker-Planck方程：质量与位置有关的不均匀介质

我们通过使用Langevin方程，在任意变形空间的广义变形导数的情况下，利用Langevin方程，提出了具有随位置变化的质量粒子的非均匀介质的Fokker-Planck方程（FPE），其中FPE与使用的变形线性（非线性）导数相关。具有与位置有关的扩散系数的不均匀介质的FPE等效于变形空间内的变形FPE（用广义导数描述）和恒定扩散系数。当温度和迁移率具有相同的位置相关函数形式时，变形的FPE与非均匀介质的扩散方程式一致，并且与非线性Langevin方法一致。的变形版本$H$-定理允许将Boltzmann-Gibbs熵函数表示为两种贡献的总和，一种来自粒子，另一种来自非均匀介质。形式主义用无限方井和具有线性漂移系数的约束势来说明。还讨论了超统计量与位置相关的Langevin方程之间的联系。