Self-organized escape of oscillator chains in nonlinear potentials

We present the noise-free escape of a chain of linearly interacting units from a metastable state over a cubic on-site potential barrier. The underlying dynamics is conservative and purely deterministic. The mutual interplay between nonlinearity and harmonic interactions causes an initially uniform lattice state to become unstable, leading to an energy redistribution with strong localization. As a result, a spontaneously emerging localized mode grows into a critical nucleus. By surpassing this transition state, the nonlinear chain manages a self-organized, deterministic barrier crossing. Most strikingly, these noise-free, collective nonlinear escape events proceed generally by far faster than transitions assisted by thermal noise when the ratio between the average energy supplied per unit in the chain and the potential barrier energy assumes small values.

Figure 1

Potential barrier landscape with a chain positioned at the bottom. The parameter values are $a=1$ and ${\omega }_0^2=2$.

Figure 2

Growth rate in dependence of the wave number $Q$ for different coupling strengths $\kappa$, as indicated on the graphs. The parameter values are $k=0$, $u_0=0.2$, $a=1$, ${\omega }_0^2=2$, and $N=100$.

Figure 3

Mean growth rate versus the coupling strength. The parameter values used are the same as in Fig. 2.

Figure 4

Spatiotemporal evolution of the energy distribution $E_n\left(t\right)$. Initially, the coordinates are uniformly distributed $\left(k=0\right)$ within the interval $\left|q_n\left(0\right)-q_0\right|\le \Delta q$ with an average at $q_0=0.3$ and width $\Delta q=0.01$ and zero momenta—i.e., $p_0=\Delta p=0$. This yields a total energy $E_{total}=8.1\equiv 6.075\Delta E$. The parameter values are given by $a=1$, ${\omega }_0^2=2$, $N=100$, and $\kappa =0.3$.

Figure 5

Snapshots of the amplitudes $q_n\left(t\right)$ of a segment of the lattice covering ten oscillators, illustrating the formation of a localized structure with a subsequent barrier crossing event of the chain. Initial conditions are $q_0=0.45$, $\Delta q=0.01$, and $p_0=\Delta p=0$. Parameter values are ${\omega }_0^2=2$, $a=1$, $N=100$, and $\kappa =0.2$. Top panel: snapshot taken at time $t=473$: Emergence of localized structures. Bottom panel: snapshot taken at time $t=673$: Depicted is a segment of the lattice with one unit located beyond the barrier. The dashed lines in the $\left(q_n-n\right)$ plane designate the position of the potential’s minimum and maximum.

Figure 6

First intersections of the stable $W_s$ (thin gray line) and the unstable $W_u$ manifold (thick dark line) of the hyperbolic point of the map in Eq. (17) at the map origin (0, 0). The parameter values are ${\omega }_0^2=2$, $a=1$, and $\kappa =1$. The homoclinic point labeled $\left({\tilde{q}}_{n_c}^h;{\tilde{q}}_{n_c+1}^h\right)$ and its map iterate $\left({\tilde{q}}_{n_c+1}^h;{\tilde{q}}_{n_c+2}^h\right)$ correspond on the consecutive lattice sites $n=n_c$, $n_c+1$, $n_c+2$ to a decaying pattern whose amplitudes with ${\tilde{q}}_n^h>{\tilde{q}}_{n+1}^h$ are represented by the arrows of varying length in the schematic representation in the upper left corner. Map iterations of the homoclinic point $\left({\tilde{q}}_{n_c+1}^h;{\tilde{q}}_{n_c+2}^h\right)$ result in further homoclinic points approaching asymptotically the hyperbolic point at the map origin.

Figure 7

Amplitude profile of the critical chain configuration, being symmetrically centered at site $n_c$, for different coupling strengths $\kappa =0.1$ (circles), $\kappa =0.3$ (diamonds), and $\kappa =1.0$ (triangles). For better illustration only we depict the central part of the lattice chain with sizable elongations. The parameter values are ${\omega }_0^2=2$ and $a=1$.

Figure 8

Activation energy $E_{act}$ as a function of the coupling strength $\kappa$. Solid line: fitted data with slope $s=3.54\pm 0.01$. The remaining parameter values are ${\omega }_0^2=2$ and $a=1$.

Figure 9

The escape time $T_{esc}^{\left(n\right)}$ of the individual chain units versus position for collective escape and periodic boundary conditions for a chain consisting of 100 units for one realization of initial conditions. The parameter values are $a=1$, ${\omega }_0^2=2$, $N=100$, and $\kappa =0.3$. The initial conditions are given by $q_0=0.45$, $\Delta q=0.1$, and $p_0=\Delta p=0$.

Figure 10

Mean escape time $T_{esc}$ vs the ratio $E_0∕\Delta E$ of energy per unit $E_0$ and the barrier energy $\Delta E$ for the noise-free case with initial $k=0$ mode (solid line) with $\Delta q=0.01$ and $p_0=\Delta p=0$ (for $q_0$, see text), the case with strongly randomized initial conditions (diamonds) where $-0.251\le q_n\left(0\right)\le 0.534$, $-0.251\le p_n\left(0\right)\le 0.534$, and the noise-assisted escape (dashed line), respectively. The parameter values are ${\omega }_0^2=2$, $a=1$, and $\kappa =0.3$ and the friction strength is $\nu =0.1$. The inset depicts the same data, but now plotted versus the effective Arrhenius factor $\Delta E∕E_0$.

Figure 11

Resonant behavior in the mean escape time $T_{esc}$ as a function of the coupling strength $\kappa$ with the remaining parameter values as in Fig. 4. The initial conditions are for the coordinates $q_0=0.45$ and $\Delta q=0.1$ and for the momenta $p_0=\Delta p=0$. The averages were performed over 500 realizations of random initial conditions.

Figure 12

Spatially localized structure at $t=500$ for three different coupling strengths. Initial conditions and parameter values as in Fig. 4 except for the coupling strengths (a) $\kappa =0.05$, (b) $\kappa =0.31$, and (c) $\kappa =1.00$.

Figure 13

The ratio $\chi$ defined in Eq. (31) as a function of the coupling strength $\kappa$. The parameter values are ${\omega }_0^2=2$ and $a=1$. The initial conditions are given by $q_0=0.45$ and $p_0=0$.

Figure 14

First, mean, and last escape time $T_{esc}$ as a function of the chain length $N$. The parameter values are ${\omega }_0^2=2$, $a=1$, and $\kappa =0.2$. The initial conditions are given by $q_0=0.50$, $\Delta q=0.01$, and $p_0=\Delta p=0$.

Figure 15

Mean escape time $T_{ecs}$ as a function of the number of oscillators, $N$, for different values of $E_{total}$. For the choice of $q_0$, see text. Furthermore, $\Delta q=0.01$ and $p_0=\Delta p=0$. The parameter values are ${\omega }_0^2=2$, $a=1$, $\kappa =0.2$, and $E_{total}$ as depicted in the legend.