Nonlinear response of a linear chain to weak driving

We study the escape of a chain of coupled units over the barrier of a metastable potential. It is demonstrated that a very weak external driving field with a suitably chosen frequency suffices to accomplish speedy escape. The latter requires passage through a transition state, the formation of which is triggered by permanent feeding of energy from a phonon background into humps of localized energy and elastic interaction of the arising breather solutions. In fact, cooperativity between the units of the chain entailing coordinated energy transfer is shown to be crucial for enhancing the rate of escape in an extremely effective and low-energy cost way where the effects of entropic localization and breather coalescence conspire.

Figure 1

Time evolution of the spatial Fourier transform of $q_n\left(t\right)$. The values of the parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, $\omega =1.4$, ${\theta }_0=0$, and $\kappa =0.1$.

Figure 2

Spatiotemporal pattern of the coordinates $q_n\left(t\right)$ prior to escape for very weak periodic driving. The values of the parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, $\omega =1.4$, ${\theta }_0=0$, and $\kappa =0.1$. The pattern is shown up to times shortly before the escape of the chain takes place.

Figure 3

Temporal evolution of the average kinetic (solid line) and average potential (dashed line) energy for weak periodic driving for coupled units ($\kappa =0.1$) and uncoupled units ($\kappa =0$) as indicated in the plot. The values of the remaining parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, $\omega =1.4$, and ${\theta }_0=0$. The time evolution is shown up to times shortly before the escape of the chain takes place.

Figure 4

Temperature of the chain for weak periodic driving. The values of the parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, $\omega =1.4$, ${\theta }_0=0$, and $\kappa =0.1$. The time evolution is shown up to times shortly before the escape of the chain takes place.

Figure 5

Work done on the chain by the weak periodic driving for $\kappa =0$ (dashed line) and $\kappa =0.1$ (solid line). The values of the remaining parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, $\omega =1.4$, and ${\theta }_0=0$.

Figure 6

Mean escape time of the chain as a function of the driving frequency $\omega$ for a small fixed driving strength $A=0.003$. The values of the remaining parameters are ${\omega }_0^2=2$, $a=1$, $A=0.003$, ${\theta }_0=0$, and $\kappa =0.1$.