Roughness-enhanced transport in a tilted ratchet driven by Lévy noise

The enhanced transport of particles by roughness in a tilted rough ratchet potential subject to a Lévy noise is investigated in this paper. Due to the roughness, the transport process exhibits quite different properties compared to the smooth case. We find that the roughness on the potential wall functions like a ladder to provide the convenience for particles to climb up but hinder them to slide down. The mean first passage time from one well to its right adjacent well and the mean velocity are, respectively, calculated versus the roughness, the external force, and the Lévy stability index. Our results show that the roughness is able to induce an enhancement on the mean velocity of particles and accelerate the barrier crossing process. The general conditions require a small external force and a small Lévy stability index. We find that with increasing external forces, the enhancement areas of roughness and Lévy stability index both shrink. However, for the Lévy stability index within the enhancement area, its increase will enlarge the enhancement area of roughness. On the contrary, under the same conditions we observe that for a Gaussian noise the roughness always reduces the corresponding mean velocity which is very different from the case of Lévy noise.

Figure 1

An illustration of the effective rough potential $U_{\mathrm{eff}}\left(x\right)$ in Eq. (3). The period length of $U_0\left(x\right)$ is taken as $L=1.0$, and the external force $F=2.0$. The two labeled stable points of $U_0\left(x\right)-Fx$ are $x_0=\arcsin \left(F/2\pi \right)/2\pi$ and $x_L=L+x_0$.

Figure 2

MVs with respect to the external force $F$ and roughness $\varepsilon$ for different $\alpha$, with $D=0.3$. (a) $\alpha =1.85$; roughness can increase the MV for small $F$. (b) $\alpha =2.0$; roughness always decreases the MV. (c) $\alpha =1.6$; the enhancement areas of $\varepsilon$ shrink with increasing $F$. (d) $\alpha =1.9$; the influence of $\varepsilon$ suddenly changes from enhancement to suppression with increasing $F$. (e) The space diagram of MV for $\alpha =1.5$. The inset shows the enhancement area of $\varepsilon$, in which the difference $\mathrm{\Delta }\varepsilon =0.02$ makes the changes of bar unsmooth. (f) is the vertical view of $v_{\varepsilon }-v_{\varepsilon =0}$, where we can find the enhancement area and optimal $\varepsilon$ for different $F$. The legend in (a) and (b) for $\varepsilon$ holds throughout the paper.

Figure 3

MVs as a function of $\alpha$ for different $F$ with $D=0.3$. The shaded parts are enhancement areas of $\alpha$ in which roughness increases MVs. The enhancement areas shrink with increasing $F$. The insets in (c) and (d) are plotted to illustrate that small $\alpha$ is within the enhancement areas for some $\varepsilon <0.1$. (e) The bifurcation diagram of $\alpha$ and $F$, in which the gray color corresponds to the enhancement area, and the white is the suppressing area.

Figure 4

(a), (b) MVs as a function of $\varepsilon$ for different $\alpha$ with $D=0.3$ in cases $F=2.0$ and $F=4.0$. With increasing $\alpha$, the enhancement areas of $\varepsilon$ increase for small $\alpha$, but then encounter a sudden change from enhancement to suppression. (c) The space diagram of MV with respect to $\varepsilon$ and $\alpha$ for $F=2.0$. (d) The vertical view of $v_{\varepsilon }-v_{\varepsilon =0}$, from which we can find the variation of optimal $\varepsilon$.

Figure 5

MFPTs with respect to $F$ with $D=0.3$. For $\alpha =1.9$ the roughness can decrease the MFPT for small $F$; however, for $\alpha =1.98$ the roughness always increases the MFPT.

Figure 6

MFPTs with respect to $\alpha$ and $\varepsilon$ with $D=0.3$. With increasing $F$, the enhancement areas of $\alpha$ shrink. The insets in (a)–( c) illustrate that $\alpha =1.2$ decreases MFPTs for some $\varepsilon <0.1$. (d) shows the influences of $\varepsilon$ for different $\alpha$.

Figure 7

MFPTs with respect to $D$ and $\varepsilon$, for $\alpha =1.8,F=2.0$.