Transition between locked and running states for dimer motion induced by periodic external driving

We study the motion of a dimer in a one-dimensional spatially periodic washboard potential. The tilt of the latter is time-periodically modulated by an ac field. We focus interest on the detrapping of the (static) ground state solution of the dimer caused by the ac field. Moreover, we demonstrate that slow tilt modulations not only induce a trapping-detrapping transition but drive the dimer dynamics into a regime of transient long-range running states. Most strikingly, the motion proceeds then unidirectionally, so that the dimer covers huge distances regardless of the fact that the bias force in the driven system vanishes on the average. We elucidate the underlying dynamics in phase space and associate long-lasting running states with the motion in ballistic channels occurring due to stickiness to invariant tori.

Figure 1

Barrier crossing of the trajectory of the dimer emanating from the minimum, labeled as $M$, in one potential well and passing through a saddle, labeled as $S$, represented by the wiggling solid line. The untilted potential energy landscape $U_{\text{eff}}\left(Q_x,Q_y\right)$ is additionally depicted. The parameter values are $l_0=0.5$, $\kappa =5$, $F=0.4$, $\Omega ={10}^{-3}$, and ${\Theta }_0=0$.

Figure 2

(Color online) Potential energy $U_{\text{eff}}$ as a function of the difference coordinate $Q_y$ for fixed $Q_{x,g}=0$ (labeled 1 and 3 for coupling strength $\kappa =0.5$ and 5, respectively) and fixed $Q_{x,s}=0.25$ (labeled 2 and 4) for coupling strength $\kappa =0.5$ and 5, respectively). The minimum of the graphs 1 and 3 determines the position of the ground state $Q_{y,g}$ while for the graphs 2 and 4 the minimum lies at the value of the saddle point $Q_{y,s}=-l_0/2=-0.25$. The remaining parameter values are $l_0=0.5$ and $F=0$.

Figure 3

(Color online) Critical coupling strength for dynamical detrapping as a function of the equilibrium length. The parameter values are $F=0.4$ and ${\Theta }_0=0$. For the choice of $\Omega$ we refer to the text.

Figure 4

(Color online) Time evolution of the center of mass coordinate $Q_x$. Left panel: Fixed coupling strength $\kappa =5$ and various modulation frequencies as indicated in the plot. Right panel: Fixed $\Omega ={10}^{-3}$ and different values of $\kappa$ as indicated in the plot. The remaining parameter values are $l_0=0.5$ and $F=0.4$.

Figure 5

(Color online) Time evolution of the difference coordinate $Q_y$. The parameter values are $l_0=0.5$, $\kappa =5$, $F=0.4$, $\Omega ={10}^{-3}$, and ${\Theta }_0=0$.

Figure 6

(Color online) Poincaré plots presented in the $\Theta \text{−}{P}_x$ plane. The phase variable $\Theta$ is given $\text{mod}\left(2\pi \right)$. Illustrated are a driving with $\Omega =1$ (top panel) and an adiabatic driving with $\Omega ={10}^{-3}$ (bottom panel). The remaining parameter values are $l_0=0.5$, $\kappa =0.5$, $F=0.4$, and ${\Theta }_0=0$.