# Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction

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## Abstract

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## 1. Introduction

## 2. The Non-Linear Fokker–Planck Equation with Power-Law Diffusion

## 3. Direction-Dependent Drag Forces and Non-Linear Fokker–Planck Equations

## 4. Stationary Solutions and $\mathit{H}$-Theorem for Over-Damped Systems with Position-Dependent Drag forces

#### 4.1. Stationary Solutions

#### 4.2. H-Theorem

## 5. An Example with a Time-Dependent Solution Having Asymmetric $\mathit{q}$-Gaussian Form

## 6. Exact Solutions for Non-Linear Diffusion with General $\mathit{q}$-Values

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the asymmetric, q-Gaussian solution (30) to the non-linear diffusion Equation (27). The density $\rho (x,t)$ is depicted as a function of x for different values of $t/\tau $. The solution corresponds to $D=1$ and $q=0$, and to initial conditions given by $A\left(0\right)=1$, ${\beta}_{1}\left(0\right)=1$, ${\beta}_{2}\left(0\right)=1/4$. The density $\rho $ is measured in units of $A\left(0\right)$, and the coordinate x in units of ${\beta}_{1}{\left(0\right)}^{-1/2}$.

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**MDPI and ACS Style**

Plastino, A.R.; Wedemann, R.S.; Tsallis, C.
Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction. *Symmetry* **2021**, *13*, 1621.
https://doi.org/10.3390/sym13091621

**AMA Style**

Plastino AR, Wedemann RS, Tsallis C.
Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction. *Symmetry*. 2021; 13(9):1621.
https://doi.org/10.3390/sym13091621

**Chicago/Turabian Style**

Plastino, Angel Ricardo, Roseli S. Wedemann, and Constantino Tsallis.
2021. "Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction" *Symmetry* 13, no. 9: 1621.
https://doi.org/10.3390/sym13091621