Roughness in the periodic potential enhances transport in a driven inertial ratchet

We study the effects of roughness in the asymmetric periodic potential on the transport and diffusion of an inertial Brownian particle driven by a time-periodic force in a Gaussian environment. We find that moderate roughness leads to the loss of transient anomalous diffusion, and it helps to establish normal diffusion in the weak noise limit. We uncover a contrasting effect of roughness on the transport of particles in the weak and moderate to large noise limit. In the weak noise limit, small amplitude roughness results in the increase of directed transport, whereas in the moderate to large noise limit, roughness hinders transport. The deterministic dynamics of the system reveals that the purely periodic system under smooth potential transits into a chaotic system due to the moderate roughness in the potential. Therefore our calculations demonstrate the constructive role of roughness in the transport of particles in the inertial regime.

Figure 1

Sketch of the asymmetric periodic potential in Eq. (1), $U\left(x\right)$, without ($\epsilon =0$) and with roughness ($\epsilon =0.1$).

Figure 2

Time dependence of the diffusion coefficient, $D\left(t\right)$, for the different values of roughness $\epsilon$ and noise strength $Q$.

Figure 3

Variation of the maximum value of the time-dependent diffusion coefficient, $D{\left(t\right)}_{\max }$, as a function of $\epsilon$ for different values of $Q$.

Figure 4

Ensemble- and period-averaged velocity $v\left(t\right)$ as a function of time $t$ for different values of roughness $\epsilon$ and noise strength $Q$.

Figure 5

Ensemble- and period-averaged asymptotic velocity $\langle v\rangle$ as a function of noise strength $Q$ for different values of $\epsilon$. The vertical dashed line separates the lower and higher noise regimes.

Figure 6

The dependence of the ensemble- and period-averaged asymptotic velocity $\langle v\rangle$ as a function of $\epsilon$ for different values of $Q$. Left and right columns correspond to weak and strong noise limits, respectively.

Figure 7

Basins of attraction of asymptotic velocity $\langle v\rangle$ from the deterministic dynamics for different values of roughness $\epsilon$. The color bar at the top represents the magnitude of asymptotic velocity $\langle v\rangle$.

Figure 8

Comparison of asymptotic velocity $\langle v\rangle$ with $\epsilon$ from the deterministic (top) and stochastic systems with weak noise strength $Q=0.0001$ (bottom) for the different values of asymmetric parameter $\psi$.

Figure 9

The time series of position (first row) and velocity (second row), phase-space map (third row), and power spectra of velocity (fourth row) in the absence of noise $Q=0$ for varying amplitudes of roughness $\epsilon$. The initial value chosen for position and velocity are 0.1 and 0, respectively, for the single trajectory.

Figure 10

Bifurcation plot as a function of amplitude of roughness $\epsilon$ for the deterministic system.

Figure 11

The dependence of the ensemble- and period-averaged asymptotic velocity ${\langle v\rangle }_d$ as a function of $\epsilon$ for the different values of particle mass and $\psi$.