# Diffusion Properties of a Brownian Ratchet with Coulomb Friction

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## Abstract

**:**

## 1. Introduction

## 2. Langevin Equation with Coulomb Friction and Ratchet Effect

## 3. Diffusion Properties

## 4. Constant Force Model

#### Characteristic Escape Times from a Single Well

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**

**Average ratchet drift vs. temperature T and friction coefficient $\gamma $.**(

**a**): Contour plot for the average ratchet velocity $\langle v\left(t\right)\rangle $ as a function of the temperature T and the friction $\gamma $. Parameters: $\alpha =1.0$, $\mu =0.4$. The blue horizontal and vertical lines highlight the $\langle v\left(t\right)\rangle $ values at fixed T and $\gamma $ used to plot panels (

**b**) and (

**c**), respectively. (

**b**) and (

**c**): Trend of $\langle v\left(t\right)\rangle $ as a function of $\gamma $ at fixed $T=10.0$ and T at fixed $\gamma =0.05$, respectively, as extracted from panel (

**a**).

**Figure 2.**

**Variance vs. temperature T and friction coefficient $\gamma $.**Position variance $Var\left[x\right(t\left)\right]$ for various choices of temperature T at fixed $\gamma =0.05$ (panel (

**a**)) and for various choices of friction coefficient $\gamma $ at fixed $T=10$ (panel (

**b**)) up to simulation time $t={10}^{6}$. The insets report the ratio between the measured diffusion coefficient ${D}_{m}$ and the asymptotic underdamped one $D=T/\gamma $. Other parameters: $\alpha =1.0$, $\mu =0.4$.

**Figure 3.**

**Diffusion properties vs. temperature T and friction coefficient $\gamma $.**(

**a**) and (

**c**): Position mean square displacement $\Delta (t,0)$ for various choices of temperature T at fixed $\gamma =0.05$ (panel (

**a**)) and friction coefficient $\gamma $ at fixed $T=10.0$ (panel (

**c**)) up to simulation time $t={10}^{6}$. (

**b**) and (

**d**): Diffusive→ballistic crossover times ${t}_{db}$ as a function of the temperature T at fixed $\gamma =0.05$ (panel (

**b**)) and friction coefficient $\gamma $ at fixed $T=10.0$ (panel (

**d**)) measured from $\Delta (t,0)$. The insets report the measured ballistic→diffusive crossover times ${t}_{bd}$. Other parameters: $\alpha =1.0$, $\mu =0.4$.

**Figure 4.**

**Constant-force model.**(

**a**): Mean square displacement $\Delta (t,0)$ of the constant force model Equation (3) compared to those computed in the no-potential ($U=0$ in Equation (1)), no-Coulomb friction ($\alpha =0$ in Equation (1)), and ratchet model (Equation (1) with $\alpha =1$). Points denote numerical results, and the blue solid line is the theoretical expression Equation (5). The inset reports the theoretical and numerical $\Delta (t,0)$ for a different force, highlighting the $\sim {t}^{3}$ superdiffusive regime. (

**b**) and (

**c**): Comparison between the computed ${t}_{db}$ in the ratchet and in the constant force model as a function of the temperature T and of the friction coefficient $\gamma $, respectively. The blue dots are evaluated through (6), and red dots are numerically estimated. Parameters: $\alpha =1.0$, $\mu =0.4$, $\gamma =10.0$ in panel (

**b**); $T=0.05$ in panel (

**c**). $F=2.28\xb7{10}^{-4}$ is the constant force corresponding to the $T=10,\gamma =0.05,\mu =0.4$ ratchet case, while $F=1.0,{v}_{0}=1.0$ are the parameters chosen for the inset.

**Figure 5.**

**Average escape times.**Average escape time $\langle {t}_{e}\rangle $ for a harmonically confined Brownian particle with and without Coulomb friction and constant-force starting at ${x}_{0}=0$ as a function of the right escape point. The inset reports a graphical depiction of the harmonic approximation introduced in the main text. For the sake of clarity, the harmonic potential is horizontally and graphically shifted. Parameters: $\gamma =0.05$, $T=10.0$, ($k=2.20$) ($\alpha =1.0$). ${d}_{0}=1.49$ is chosen in such a way that $k{d}_{0}^{2}/2=\Delta U$, with $\Delta U$ as the ratchet potential depth for $\mu =0.4$. $F=2.28\times {10}^{-4}$ is the constant force corresponding to the $\gamma =0.05,T=10,\mu =0.4$ ratchet case.

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**MDPI and ACS Style**

Semeraro, M.; Gonnella, G.; Lippiello, E.; Sarracino, A.
Diffusion Properties of a Brownian Ratchet with Coulomb Friction. *Symmetry* **2023**, *15*, 200.
https://doi.org/10.3390/sym15010200

**AMA Style**

Semeraro M, Gonnella G, Lippiello E, Sarracino A.
Diffusion Properties of a Brownian Ratchet with Coulomb Friction. *Symmetry*. 2023; 15(1):200.
https://doi.org/10.3390/sym15010200

**Chicago/Turabian Style**

Semeraro, Massimiliano, Giuseppe Gonnella, Eugenio Lippiello, and Alessandro Sarracino.
2023. "Diffusion Properties of a Brownian Ratchet with Coulomb Friction" *Symmetry* 15, no. 1: 200.
https://doi.org/10.3390/sym15010200