Nonsinusoidal current and current reversals in a gating ratchet

In this work, the ratchet dynamics of Brownian particles driven by an external sinusoidal (harmonic) force is investigated. The gating ratchet effect is observed when another harmonic is used to modulate the spatially symmetric potential in which the particles move. For small amplitudes of the harmonics, it is shown that the current (average velocity) of particles exhibits a sinusoidal shape as a function of a precise combination of the phases of both harmonics. By increasing the amplitudes of the harmonics beyond the small-limit regime, departures from the sinusoidal behavior are observed and current reversals can also be induced. These current reversals persist even for the overdamped dynamics of the particles.

Figure 1

$v$ vs $\varphi$ from simulations of (7). Filled circles with error bars: current in steady state computed from the Eq. (13). Solid line represents the fitting curve of the circles, $v=-0.00170cos(\varphi +2.195)$. Parameters: $m=1$, $\alpha =1$, $U_0=5$, ${\epsilon }_1={\epsilon }_2=0.5$, $\omega =1$, $q_1=1$, $q_2=2$, and $D=1$. Dotted line represents zero velocity.

Figure 2

$v$ vs $\varphi$ from simulations of (7) shows nonsinusoidal behavior (filled circles with error bars). Solid line represents the fitting curve $v=-0.4168cos(\varphi +0.7713)-0.0686cos(3\varphi -0.0909)$. Parameters: $m=1$, $\alpha =1$, $U_0=5$, ${\epsilon }_1={\epsilon }_2=2$, $\omega =1$, $q_1=1$, $q_2=2$, and $D=1$. Final time of integration 1000.

Figure 3

$v$ vs $\epsilon$ from simulations of (7). Filled circles: current from the Eq. (13). Dashed line: fitting curve $v=-0.0207{\epsilon }^2(1-\epsilon )$, predicted for small-amplitude limit. Solid line represents the fitting curve of the simulation points predicted in the intermediate amplitude regime, $v={\sum }_{k=0}^5{a}_k{\epsilon }^{k+2}$, with $a_0=-0.002$, $a_1=-0.076$, $a_2=0.367$, $a_3=-0.747$, $a_4=0.664$, $a_5=-0.205$. Parameters: $m=1$, $\alpha =1$, $U_0=5$, ${\epsilon }_1=A\epsilon$, ${\epsilon }_2=A(1-\epsilon )$, $A=1$, $\varphi =4.09$, $\omega =1$, $q_1=1$, $q_2=2$, and $D=1$.

Figure 4

$v$ vs $\epsilon$ from simulations of (7) shows that the direction of the current changes at approximately $\epsilon =0.5$. Parameters: $m=1$, $\alpha =1$, $U_0=5$, ${\epsilon }_1=2\epsilon$, ${\epsilon }_2=2(1-\epsilon )$, $\omega =1$, $\varphi =2.8$, $q_1=1$, $q_2=2$, and $D=1$. Final time of integration 1000. Dotted line represents zero velocity.

Figure 5

$v$ vs $\varphi$ from simulations of the overdamped system (11) (filled circles with error bars). Solid line represents the fitting curve $v=0.0033cos\left(\varphi \right)$. Parameters: $\alpha =1$, $U_0=5$, ${\epsilon }_1={\epsilon }_2=0.5$, $\omega =1$, $q_1=1$, $q_2=2$, and $D=1$.

Figure 6

$v$ vs $\varphi$ from simulations of the overdamped system (11), filled circles with error bars. Dashed and solid lines are the fitting curves $v=-0.345cos\left(\varphi \right)$ and $v=-0.344cos\left(\varphi \right)-0.042cos\left(3\varphi \right)$, respectively. Parameters: $\alpha =1$, $U_0=5$, ${\epsilon }_1={\epsilon }_2=2$, $\omega =1$, $q_1=1$, $q_2=2$, and $D=1$.

Figure 7

$v$ vs $\epsilon$ from simulations of (11) shows that the direction of the current changes around $\epsilon \approx 0.75$. Amplitudes of $f_1$ and $f_2$ are varied simultaneously as ${\epsilon }_1=2\epsilon$, ${\epsilon }_2=2\epsilon$. The rest of parameters are: $\alpha =1$, $U_0=5$, $\omega =1$, $q_1=1$, $q_2=2$, $\varphi =2.8$, and $D=1$. 20000 realizations.

Figure 8

$v$ vs $\epsilon$ from simulations of (11) shows that the direction of the current also changes around $\epsilon \approx 0.75$ (filled circles) when the amplitudes are varied with constant total amplitude 2 as ${\epsilon }_1=2\epsilon$, ${\epsilon }_2=2(1-\epsilon )$. The rest of parameters are $\alpha =1$, $U_0=5$, $\omega =1$, $q_1=1$, $q_2=2$, $\varphi =2.8$, and $D=1$. 20000 realizations.

Figure 9

Contour plots $v$ as functions of ${\epsilon }_1$ and ${\epsilon }_2$ from simulations of (11). The thick blue line marks the inversion of the current. Left: $U_0=5$. Right: $U_0=2.5$. The rest of parameters are $\alpha =1$, $\omega =1$, $q_1=1$, $q_2=2$, $\varphi =2.8$, and $D=1$.