Dynamics below the Depinning Threshold in Disordered Elastic Systems

We study the steady-state low-temperature dynamics of an elastic line in a disordered medium below the depinning threshold. Analogously to the equilibrium dynamics, in the limit $T\to 0$, the steady state is dominated by a single configuration which is occupied with probability 1. We develop an exact algorithm to target this dominant configuration and to analyze its geometrical properties as a function of the driving force. The roughness exponent of the line at large scales is identical to the one at depinning. No length scale diverges in the steady-state regime as the depinning threshold is approached from below. We do find a divergent length, but it is associated only with the transient relaxation between metastable states.

Figure 1

Dynamical phase diagram. The steady-state properties of the elastic string at $T\to 0$ are determined by $L_{\mathrm{opt}}$ (solid symbols) and $\xi$. They separate regions characterized by the equilibrium exponent, the depinning exponent (gray region), and the thermal one. The divergent length $L_{\mathrm{relax}}$ (open symbols) is associated only with transient dynamics. Lines are guides to the eye.

Figure 2

Low-temperature dynamics of the driven elastic string below the depinning threshold: the optimal path to escape from a given metastable configuration $\alpha$ pass through a saddle configuration $\beta$ that can relax deterministically to the next metastable configuration $\gamma$ with $E_{\gamma }<E_{\alpha }$. Configurations $\alpha$ and $\beta$ differ on a length $L_{\mathrm{opt}}$; $\alpha$ and $\gamma$ differ on a length $L_{\mathrm{relax}}$.

Figure 3

Steady-state structure factor of the line in the $T\to 0$ limit, averaged on 1000 samples. (a) $S(q)$ for $L=32$, $M=92$, and different forces (curves are shifted for clarity). (b) $S(q)$ at $f/f_c=0.8$ for $L=16$, 32, 64, 128, and $M=L^{{\zeta }_{\mathrm{dep}}}$.