Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass

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.

Figure 1

2D (upper line) and 3D (bottom line) representations of the solutions $P(x,t)$ of the inhomogeneous FPE for free particle with parameters ${\gamma }_q{l}_0=0$ (usual case), 0.2 and 0.4. An asymmetry with respect to $x=0$ increases with ${\gamma }_q{l}_0$. The divergence of Eq. (22) for $x\to x_d=-1/{\gamma }_q$ implies a diffusion limited to the interval $x<x_d$. Notice that the space and time axis of the 3D plots are inverted (increase from right to left), for a better visualization.

Figure 2

3D representation of $P(x,t)$ for ${\gamma }_q{l}_0={10}^2$ showing that diffusion is stopped at $x=0$ for sufficiently high values of ${\gamma }_q{l}_0$ (illustrated with $x_d=-{10}^{-2})$ for a better visualization.

Figure 3

Plot of $\langle {\left(\mathrm{\Delta }x\right)}^2\left(t\right)\rangle =\langle x^2\left(t\right)\rangle -{\langle x\left(t\right)\rangle }^2$ as a function of time for a free particle. Normal diffusion behavior, $\langle {\left(\mathrm{\Delta }x\right)}^2\rangle \approx \mathrm{\Gamma }t$, is observed for $t/\tau \ll 1, \forall {\gamma }_q{l}_0$, and exponential hyperdiffusion, $\langle {\left(\mathrm{\Delta }x\right)}^2\rangle \propto e^{2t/\tau }$, for $t/\tau \gg 1$.

Figure 4

FPE for an inhomogeneous media with a linear potential. (a) Stationary solution for different values of ${\sigma }_0{\gamma }_q$. Similarly to the free particle (see Fig. 1), asymmetry in the PDF is observed. (b) Entropy $\mathcal{S}$ of the system (black line), the contribution of the entropy of the particles $S_{\text{BG}}$ (blue dashed line), and the contribution of the medium, $-\langle ln(1+{\gamma }_{q}x)\rangle$, (red dash-dot line), (see Eq. (66)), as a function of the inhomogeneity of the medium, controlled by ${\gamma }_q{\sigma }_0$.