Spatiotemporal scaling for out-of-equilibrium relaxation dynamics of an elastic manifold in random media: Crossover between diffusive and glassy regimes

We study the relaxation dynamics of a three-dimensional elastic manifold in a random potential from a uniform initial condition by numerically solving the Langevin equation. We observe the growth of the roughness of the system up to larger wavelengths with time. We analyze the structure factor in detail and find a compact scaling ansatz describing two distinct time regimes and the crossover between them. We find that a short-time regime corresponding to a length scale smaller than the Larkin length $L_c$ is well described by the Larkin model, which predicts a power-law growth of the domain size $L\left(t\right)$. Longer-time behavior exhibits a glassy regime with slower growth of $L\left(t\right)$.

Figure 1

Snapshots of the profile of the structure factor $B\left(q,t\right)$. The time changes uniformly in logarithmic scale as $t/0.34=4^0,4^1,\dots ,4^6$ from the bottom to the top. The symbols with error bars are data for $L_{\text{sys}}=128$, and dotted lines are for $L_{\text{sys}}=256$. The arrows connect the values in the Larkin model (bold curves) given by Eq. (6) and numerical data obtained in the sinusoidal model at the same time $t$. The vertical line shows $q^2=t_c^{-1}$ (Ref. 47).

Figure 2

(Color online) Time evolution of $\sigma \left(t\right)$. The inset is a semilogarithmic plot of the same data. The solid line curves show the results of the Larkin model for $h⪡T$. The horizontal dotted line indicates the equilibrium value $\sigma \left(t=\infty \right)$ for $h=0$. The lines denoted with $2\pi c_L$ and $t_c\left(h\right)$ indicate the crossover (see the details in Ref. 47).

Figure 3

(Color online) Result of the scaling at $T=0.5$. $F\left(q,t\right)$ is plotted as a function of $X\left(t/t_c\right)Y\left(q{L}_c\right)$ for $h=1.0$, 1.6, 2.0, 2.5, 2.7, 3.0, 3.2, 3.5, and 4.0. The white curve indicates $\left(1-e^{-XY}\right)/XY$. Here the coefficients are corrected with ${\alpha }_{T\Delta }=0.027$ as in Eq. (23) and ${\alpha }_{\Delta T}=0.22$ as in Eq. (24). The rectangles in the inset show the data ranges of our simulations in $t/t_c$ and $q^2{L}_c^2$ for $h=1.0$, 2.0, 3.0, and 4.0. The bold curve in the inset indicates $X\left(t/t_c\right)Y\left(q{L}_c\right)=1$. The space-time region above this curve is equilibrated.

Figure 4

Scaling function $X\left(\tilde{t}\right)$ and $Y\left(\tilde{q}\right)$ for several fixed temperatures. The inset in the top graph indicates the correction factor $c_L\left(T\right)$. Here we use $c_L\left(T=0\right)$ by extrapolating from finite-temperature data. More precisely, the horizontal axes in this plot, $\tilde{t}$ and $\tilde{q}$, are scaled values using $L_c\left({\Delta }_h,T\right)={\left[h{c}_L\left(1.0\right)/c_L\left(T\right)\right]}^{-2}$ instead of that in Eq. (25).