Creep Motion of an Elastic String in a Random Potential

We study the creep motion of an elastic string in a two-dimensional pinning landscape by means of Langevin dynamics simulations. We find that the velocity-force characteristics are well described by the creep formula predicted from phenomenological scaling arguments. We analyze the creep exponent $\mu$ and the roughness exponent $\zeta$. Two regimes are identified: when the temperature is larger than the strength of the disorder, we find $\mu \approx 1/4$ and $\zeta \approx 2/3$, in agreement with the quasi-equilibrium-nucleation picture of creep motion; on the contrary, when lowering the temperature enough, the values of $\mu$ and $\zeta$ increase, showing a strong violation of the latter picture.

Figure 1

VF characteristics for $R(0)=0.30$. Curves correspond to $T=0.24,0.26,\dots ,0.42$ from bottom to top. Solid lines are fits of the creep law (1) with $U_c$ and $\mu$ as fitting parameters. Contrary to the naive creep prediction, the optimal fit parameter $\mu$ is temperature dependent. The inset shows the relaxation of $V(t)$ to its stationary value $V$, corresponding to the ● symbol in the main figure.

Figure 2

(a) VF characteristics: (i) $R(0)=0.12$, $T=0.24$ (*); (ii) $R(0)=0.30$, $T=0.30$ (○). Solid lines are the fitting curves using (1). The inset shows $\log (1/V)$ vs $\beta U_c(F_c/F{)}^{\mu }$ for all $T$, $R(0)$, using their respective fitting parameters $\mu (T)$ and $U_c(T)$. (b) [(c)] Structure factor $S(q)$ at $F/F_c=0.02$, for (i) [(ii)]. Solid lines are fitting curves for small $q$s. We extract $\zeta =0.67\pm 0.05$ for (b) and $\zeta =0.9\pm 0.05$ for (c). Dashed (dotted) lines correspond to the reference value $\zeta =2/3$ (${\zeta }_T=1/2$). The vertical lines indicate the approximate location of the crossover from the thermal to the random manifold scaling.

Figure 3

(a) Roughness exponent, $\zeta (T)$, and creep exponent, $\mu (T)$, vs $T$. $\zeta (T)$ (*) and $\mu (T)$ (○) correspond to $R(0)=0.30$ and $\zeta (T)$ (△), and $\mu (T)$ (□) correspond to $R(0)=0.12$. The dashed line gives the equilibrium roughness exponent ${\zeta }_{\mathrm{e}\mathrm{q}}=2/3$, and the dotted line the purely thermal roughness ${\zeta }_T=1/2$. In the inset we show a finite-size analysis of $\mu$ for the ● symbol. (b) Effective energy barriers $U_c(T)$ vs $T$ for $R(0)=0.30$ (◇) and $R(0)=0.12$ (□).