# Symmetric Fractional Diffusion and Entropy Production

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## Abstract

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## 1. Introduction

## 2. Space Symmetric Extraordinary Differential Equation

## 3. Symmetric Solutions and Entropy

## 4. Entropy Properties of the Symmetric Regime

## 5. Kullback–Leibler Entropy

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

probability density function | |

PDE | partial differential equation |

## References

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**Figure 1.**The logarithm of the symmetric stable distributions ${P}_{\alpha}(x,1)$ plotted with respect to x. The central peak sharpens as α decreases from 2 to 0. For α small enough, ${P}_{\alpha}(x,1)>1$ around $x=0$, whereas for $\alpha =2$ (Gaussian) the graph is a simple parabola.

**Figure 2.**The Shannon entropy kernel ${S}_{\alpha}^{\prime}$ against x for $\alpha =0.01,0.1,0.5$ and $t=1$. Cases where $\alpha <1$, exhibiting the remarkable dip about the origin, are compared to the Cauchy, $\alpha =1$, and Gaussian case, $\alpha =2$, that have no central dip.

**Figure 3.**Shannon entropy kernel ${S}_{\alpha}^{\prime}$ over $log\left|x\right|$ is shown for $\alpha =0.01,0.1,0.5$ and $t=1$. Cases where $\alpha <1$ are compared to the Cauchy, $\alpha =1$, and the Gaussian case, $\alpha =2$, for contrast. For lower values of α a maximum occurs that moves towards the left and a more rapid drop of the functions for lower α can be observed. Note that it can be shown that all maxima of ${S}_{\alpha}^{\prime}$ have the same value of $1/e$.

**Figure 4.**On the left the Shannon entropy ${S}_{\alpha}(t)$ over α is shown for different values of t. For small times the entropy monotonically increases with α, whereas it is monotonically decreasing for large times within the plotting range, emphasizing the non-monotonic behavior for intermediate times (cf. (17)). On the right ${S}_{\alpha}(t)$ is given over $logt$ for different values of α showing the monotonic α-dependent increase in time.

**Figure 5.**The entropy, shown as a function of α, exhibits maxima moving to smaller values of α with increasing time.

**Figure 6.**In (

**a**,

**c**,

**e**) the Kullback with respect to a Gaussian ($K({P}_{\text{G}},{P}_{\alpha})$) and a Cauchy distribution ($K({P}_{\text{C}},{P}_{\alpha})$) for various times (${10}^{-1},{10}^{0},{10}^{1},{10}^{2},{10}^{3}$) versus α are plotted. In (

**c**,

**e**) the splitting of the α-regime $(0,2]$ into $(0,1]$ and $[1,2]$ for $K({P}_{\text{C}},{P}_{\alpha})$ is indicated by the arrangement. In (

**b**,

**d**,

**f**) $K({P}_{\text{G}},{P}_{\alpha})$ and $K({P}_{\text{C}},{P}_{\alpha})$ are plotted for various α versus $logt$ (a) $K({P}_{\text{G}},{P}_{\alpha})$ is monotonic decreasing for increasing α, showing the α-ordering of the regime. At $\alpha =2$, $K({P}_{\text{G}},{P}_{\alpha})$ vanishes for all times; (

**b**) Kullback with respect to a Gaussian preserves the α-ordering for large times as the monotonic increase for $t>10$ indicates; (

**c**,

**d**) A similar result is obtained for $K({P}_{\text{C}},{P}_{\alpha})$ in the regime $\alpha \in (0,1]$. Here, $K({P}_{\text{C}},{P}_{\alpha})$ vanishes for $\alpha =1$; (

**e**) In the regime $\alpha \in [1,2]$, $K({P}_{\text{C}},{P}_{\alpha})$ trends oppositely to $K({P}_{\text{G}},{P}_{\alpha})$. The Kullback vanishes for $\alpha =1$ and increases for increasing α; (

**f**) $K({P}_{\text{C}},{P}_{\alpha})$ for various $\alpha \in [1,2]$ over $logt$ is shown. The monotonic increase demonstrates preservation of the α-ordering in the $[1,2]$ regime.

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Prehl, J.; Boldt, F.; Hoffmann, K.H.; Essex, C. Symmetric Fractional Diffusion and Entropy Production. *Entropy* **2016**, *18*, 275.
https://doi.org/10.3390/e18070275

**AMA Style**

Prehl J, Boldt F, Hoffmann KH, Essex C. Symmetric Fractional Diffusion and Entropy Production. *Entropy*. 2016; 18(7):275.
https://doi.org/10.3390/e18070275

**Chicago/Turabian Style**

Prehl, Janett, Frank Boldt, Karl Heinz Hoffmann, and Christopher Essex. 2016. "Symmetric Fractional Diffusion and Entropy Production" *Entropy* 18, no. 7: 275.
https://doi.org/10.3390/e18070275