Nonequilibrium Relaxation of an Elastic String in a Random Potential

We study the nonequilibrium motion of an elastic string in a two dimensional pinning landscape using Langevin dynamics simulations. The relaxation of a line, initially flat, is characterized by a growing length $L(t)$ separating the equilibrated short length scales from the flat long distance geometry that keeps a memory of the initial condition. We show that, in the long time limit, $L(t)$ has a nonalgebraic growth with a universal distribution function. The distribution function of waiting times is also calculated, and related to the previous distribution. The barrier distribution is narrow enough to justify arguments based on scaling of the typical barrier.

Figure 1

(a) Typical configurations of the string for different times, at $T=0.5$. (b) Structure factor for different times at $T=0.5$, averaged over 1000 disorder realizations. The dashed line corresponds to the thermal equilibrium solution which it is reached at very long times.

Figure 2

Growing characteristic length scale $L(t)$ of a string of size $L=256$. The symbols (◇) correspond to the relaxation of the clean system and the dotted line to the analytical result. The symbols (●) correspond to the disordered case. The solid line is a fit to Eq. (4), the dashed line is a fit to the power-law growth at intermediate scales. For ${10}^4<t<{10}^6$ we get $\theta \approx 0.49$. Inset: exponent $\theta$ extracted from the fit to Eq. (4) in the time interval $t_i<t<{10}^6$. The symbols (△) correspond to a system size $L=512$, and the symbols (○) to $L=256$.

Figure 3

(a) Cumulative distribution function of the plateau value $s$ for fixed times $t$ ranging from 5 to ${10}^6$. Symbols (◇) indicate the mean value $S_t$. Inset: collapse of the cumulative distributions in the rescaled variable $s/S_t$. (b) Cumulative distribution function of the barriers $u$ for $s=10,20,40,80$. Circles (●) indicate the mean value $U_s$. Step lines are obtained from raw data, while solid lines are obtained from the rescaled cumulative distribution of $s$, (5).