# Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

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## Abstract

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## 1. Introduction

## 2. The Fokker–Planck and Langevin Equations

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The probability density function (PDF) of velocity computed by the inverse Fourier transform of Equation (6) with $\alpha =2.0$ (

**left**) and $\alpha =1.5$ (

**right**) for $t=0.1,0.5,1.0,10.0$.

**Figure 2.**The PDF of velocity computed by integration of Equation (7) with with $\alpha =1.25$ (magenta line), $\alpha =1.5$ (black line), $\alpha =1.75$ (red line), and $\alpha =2.0$ (blue line) for $D/\nu =1.0$ and $\beta =0.1$.

**Figure 3.**The PDF of velocity computed by integration of Equation (7) with $\alpha =1.25$ (magenta line), $\alpha =1.5$ (black line), $\alpha =1.75$ (red line), and $\alpha =2.0$ (blue line) for $D/\nu =1.0$ and $\beta =1.0$.

**Figure 4.**The PDF of velocity computed by integration of Equation (7) with with $\alpha =1.25$ (magenta line), $\alpha =1.5$ (red line), $\alpha =1.75$ (red line), and $\alpha =2.0$ (blue line) for $D/\nu =0.1$ and $\beta =1.0$.

**Figure 5.**The Boltzmann–Gibbs entropy and the Tsallis’ entropy as functions of the fractality index q for $D/\nu =1.0$ and $\beta =0.1$ (solid black line and dashed black line, respectively), $D/\nu =1.0$ and $\beta =1.0$ (solid red line and dashed red, respectively), $D/\nu =0.1$ and $\beta =1.0$ (solid blue line and dashed blue line, respectively).

**Figure 6.**The energy and the generalized q-energy as functions of the fractality index q for $D/\nu =1.0$ and $\beta =0.1$ (solid black line and dashed black line, respectively), $D/\nu =1.0$ and $\beta =1.0$ (solid red line and dashed red, respectively), $D/\nu =0.1$ and $\beta =1.0$ (solid blue line and dashed blue line, respectively).

**Figure 7.**The ratio of the generalized diffusion coefficient (${D}_{\alpha}$) and the Brownian diffusion coefficient as functions of the fractality index $\alpha $ for $D/\nu =1.0$ and $\beta =0.1$ (red line) and $D/\nu =0.1$ and $\beta =1.0$ (black line).

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Anderson, J.; Moradi, S.; Rafiq, T.
Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes. *Entropy* **2018**, *20*, 760.
https://doi.org/10.3390/e20100760

**AMA Style**

Anderson J, Moradi S, Rafiq T.
Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes. *Entropy*. 2018; 20(10):760.
https://doi.org/10.3390/e20100760

**Chicago/Turabian Style**

Anderson, Johan, Sara Moradi, and Tariq Rafiq.
2018. "Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes" *Entropy* 20, no. 10: 760.
https://doi.org/10.3390/e20100760