# Rectification and Non-Gaussian Diffusion in Heterogeneous Media

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## Abstract

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## 1. Introduction

## 2. Diffusion in Heterogeneous Systems

## 3. Results

#### 3.1. Diffusion in an Inhomogeneous Unbounded Medium

#### 3.2. Diffusion in a Periodic Channel

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Fluctuation relation for a particle in a constant-amplitude channel and inhomogeneous medium, ${D}_{1}\ne 0$ under a constant force $\beta {f}_{0}L=0.5$. (

**A**) χ as function of the displacement, $\Delta x$, at $t{D}_{0}/{L}^{2}=0.2$ for different values of the diffusion coefficient modulation: $\frac{{D}_{1}}{{D}_{0}}=0.125,0.25,0.5$ for dotted (brown), dashed (red), and solid (orange) lines, respectively. (

**B**) $\tilde{\chi}$ as a function of $\Delta x$ for the same parameters as in panel A, where $\delta f$ has been calculated using Equation (18). (

**C**) Ω as a function of ${D}_{1}/{D}_{0}$ for $\beta {f}_{0}L=0.5$ with $\Lambda =4$ and $t{D}_{0}/{L}^{2}=0.2$; dotted line stands for $\Omega \propto {({D}_{1}/{D}_{0})}^{4}$. (

**D**) $\delta f/{f}_{0}$ as function of ${D}_{1}/{D}_{0}$ for the same parameters as in panel C; dotted line stands for $\delta f/{f}_{0}\propto {({D}_{1}/{D}_{0})}^{2}$. (

**E**) Ω as a function of time normalized by $\tau ={L}^{2}/{D}_{0}$ with $\Lambda =4$, for ${D}_{1}/{D}_{0}=0.05,0.075,0.1,0.25,0.5,0.75$; the darker the line, the larger the ratio ${D}_{1}/{D}_{0}$. (

**F**) Ω as a function of the driving force $\beta {f}_{0}L$ for ${D}_{1}/{D}_{0}=0.5$.

**Figure 2.**Fluctuation relation for a Brownian particle in a varying-amplitude channel, ${h}_{1}\ne 0$, that induces a modulation in the effective diffusion coefficient, ${D}_{1}\ne 0$, for a constant force $\beta {f}_{0}L$. (

**A**) χ as function of the displacement, $\Delta x$, for $t{D}_{0}/{L}^{2}=2$, with $\beta {f}_{0}L=1$ and different values of the entropic barrier $\Delta S=0.1,1,10$ for dotted (brown), dashed (red), and solid (orange) line, respectively. (

**B**) $p(\Delta x)$ as function of $\Delta x$ for the same values of the parameters as in panel A. (

**C**) Ω at $t{D}_{0}/{L}^{2}=2$, as a function of $\Delta S$ for $\beta {f}_{0}L=0.1,1,10$ bigger points standing for larger $\Delta S$, with $\Lambda =4$. Inset: Ω as a function of the reduced entropic barrier $\Delta S-\Delta {S}_{0}$, with $\Delta {S}_{0}=1.5$, for the same value of the parameters in the main figure; dotted line stands for $\Omega \propto {(\Delta S-\Delta {S}_{0})}^{4}$. (

**D**) $\left|\delta f\right|/{f}_{0}$ as function of $\Delta S$ for the same value of the parameters in panel C. (

**E**) Ω as a function of time normalized by $\tau ={L}^{2}/{D}_{0}$ for $\Delta S=0.1,0.25,0.5,0.75,1,2.5,5,7.5,10$, darker lines standing for larger values of $\Delta S$. (

**F**) Ω as a function of the forcing $\beta {f}_{0}L$ for $\Delta S=0.1,0.5,1,5$ darker lines standing for higher values of $\Delta S$, being $\Lambda =4$.

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Malgaretti, P.; Pagonabarraga, I.; Rubi, J.M.
Rectification and Non-Gaussian Diffusion in Heterogeneous Media. *Entropy* **2016**, *18*, 394.
https://doi.org/10.3390/e18110394

**AMA Style**

Malgaretti P, Pagonabarraga I, Rubi JM.
Rectification and Non-Gaussian Diffusion in Heterogeneous Media. *Entropy*. 2016; 18(11):394.
https://doi.org/10.3390/e18110394

**Chicago/Turabian Style**

Malgaretti, Paolo, Ignacio Pagonabarraga, and J. Miguel Rubi.
2016. "Rectification and Non-Gaussian Diffusion in Heterogeneous Media" *Entropy* 18, no. 11: 394.
https://doi.org/10.3390/e18110394