Deterministic escape dynamics of two-dimensional coupled nonlinear oscillator chains

We consider the deterministic escape dynamics of a chain of coupled oscillators under microcanonical conditions from a metastable state over a cubic potential barrier. The underlying dynamics is conservative and noise free. We introduce a two-dimensional chain model and assume that neighboring units are coupled by Morse springs. It is found that, starting from a homogeneous lattice state, due to the nonlinearity of the external potential the system self-promotes an instability of its initial preparation and initiates complex lattice dynamics leading to the formation of localized large amplitude breathers, evolving in the direction of barrier crossing, accompanied by global oscillations of the chain transverse to the barrier. A few chain units accumulate locally sufficient energy to cross the barrier. Eventually the metastable state is left and either these particles dissociate or pull the remaining chain over the barrier. We show this escape for both linear rodlike and coil-like configurations of the chain in two dimensions.

Figure 1

Coil-like structures, $l/s<1$. Snap shots of the dynamics illustrating the formation of localized structures with a subsequent barrier crossing event in the limit of weak coupling and short bond length, for one realization of initial conditions. The initial conditions are $q_{x0}=0.065$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.248$. The remaining parameter values are $a=5$, $\kappa =0.15$, $l/s=0.05$, and $N=9$. (a) Snap shot taken at time $t=0$, initially virtually flat state. (b) Snap shot taken at time $t=210$, the chain has coiled and a localized state in the $x$ direction has formed. (c) Snap shot taken at time $t=655$, one unit has overcome the barrier level. Subsequently this unit escapes while the remaining oscillators stay captured inside the potential well (not shown here). The solid lines in the $x\text{−}y$ plane designate the position of the potential’s minimum and maximum.

Figure 2

Rod-like structures, $l/s>1$. Snap shots of the dynamics illustrating the formation of localized structures with a subsequent barrier crossing event, for one realization of initial conditions. The initial conditions are $q_{x0}=0.09$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.425$. The remaining parameter values are $a=5$, $\kappa =0.9$, $l/s=1.25$, and $N=10$. (a) Snap shot taken at time $t=0$, initially virtually flat state. (b) Snap shot taken at time $t=235$, a short wavelength amplitude pattern ($\pi$ mode) in the $x$ direction has appeared. (c) Snap shot taken at time $t=245$, one unit has overcome the barrier level. Subsequently the whole chain escapes (not shown here). The solid lines in the $x\text{−}y$ plane designate the position of the potential’s minimum and maximum.

Figure 3

Formation of localized structures. Depicted are the spatiotemporal evolutions of the amplitudes $\left\{q_{xn}\right\}$ for each one realization of initial conditions. (a) $l/s<1$, formation of an array of localized solutions (large-amplitude breathers). The initial conditions are $q_{x0}=0.04$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.104$. The remaining parameter values are $a=5$, $\kappa =0.15$, $l/s=0.05$, and $N=100$. (b) $l/s>1$, formation of an array consisting of numerous breathers. The initial conditions are $q_{x0}=-0.05$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.219$. The remaining parameter values are $a=5$, $\kappa =0.5$, $l/s=2.0$, and $N=100$.

Figure 4

Mean time $T_{\text{equ}}$ in which an equipartition of energy is reached as a function of the bond length $l$. Values of the coupling strength as depicted in the legend. The initial conditions are $q_{x0}=-0.04$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.136$. The remaining parameter values are $a=5$ and $N=100$.

Figure 5

Global oscillations in the $y$ direction. Shown is the spatiotemporal evolution of the amplitudes $\left\{q_{yn}\right\}$ for one realization of initial conditions. The amplitude values are shifted with respect to their initial values $nl$. The initial conditions are $q_{x0}=-0.05$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.219$. The remaining parameter values are $a=5$, $\kappa =0.9$, $l/s=1.25$, and $N=100$.

Figure 6

Amplitude profiles of the CLM, centered at $y=0$ for two different bond lengths: $l/s=0.75$ (solid line with square symbols) and $l/s=1.25$ (dashed line with circles). For better illustration we depict only a segment of the chain, containing 5–7 units. The parameter values are $a=5$, $N=100$, and $\kappa =0.8$. The alignment of the units along the potential minimum situated at $q_x^{\text{min}}=0$ is completely arbitrary, as long as next neighbors keep their equilibrium distance $l$. In particular, completely coiled configurations can be critical transition states.

Figure 7

Illustration of the escape process for one realization of initial conditions. The individual escape times $t_{\text{esc}}$ versus the $y$ position when the corresponding unit passes the threshold value $q_x^{\text{thresh}}$ far behind the barrier. The position of the unit which overcomes the barrier first is denoted by $y_c$, all other positions are shifted by this value. The initial conditions are $q_{x0}=-0.05$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.219$. The parameter values are $a=5$, $\kappa =0.9$, $l/s=1.25$, and $N=100$.

Figure 8

Fraction of successful exits of the entire chain as a function of the coupling strength $\kappa$. The value of the bond length $l$ is given in the legend. The initial conditions are $q_{x0}=-0.05$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.219$. The remaining parameter values are $a=5$ and $N=100$.

Figure 9

Mean escape time as a function of the bond length $l$. The initial conditions are: $q_{x0}=-0.05$ and $\Delta q_x=0.001$, yielding $E_0/\Delta E=0.219$. The remaining parameter values are $a=5$, $\kappa =0.9$, and $N=100$.

Figure 10

Distribution of the escape times. The parameter values are $a=5$, $\kappa =0.9$, $l/s=1.25$, and $N=100$. Initial conditions as in Fig. 9. The inset shows a magnification of the histogram for times $1000\le t\le 2500$. The distance between subsequent peaks corresponds to the period of the global oscillations in the $y$ direction.