Creep dynamics of elastic manifolds via exact transition pathways

We study the steady state of driven elastic strings in disordered media below the depinning threshold. In the low-temperature limit, for a fixed sample, the steady state is dominated by a single configuration, which we determine exactly from the transition pathways between metastable states. We obtain the dynamical phase diagram in this limit. At variance with a thermodynamic phase transition, the depinning transition is not associated with a divergent length scale of the steady state below threshold, but only of the transient dynamics. We discuss the distribution of barrier heights, and check the validity of the dynamic phase diagram at small but finite temperatures using Langevin simulations. The phase diagram continues to hold for broken statistical tilt symmetry. We point out the relevance of our results for experiments of creep motion in elastic interfaces.

Figure 1

(Color online) An elastic string $h\left(x\right)$ in a random medium. (a) For random-bond disorder (RB), its interaction with the impurities is local. (b) For random-field disorder (RF), the energy $E_{\text{dis}}\left[h\right]$ depends on $V\left(u,x\right)$ for all $u<h$.

Figure 2

(Color online) Velocity-force characteristics of an elastic manifold at $T=0$. The three self-affine reference states are: Equilibrium $\left(f=0\right)$; Depinning $\left(f=f_c\right)$; Fast flow $\left(f⪢f_c\right)$.

Figure 3

(a) Zero-temperature dynamical phase diagram. (b) Schematic structure factor for $f>f_c$. The correlation length $\xi$ fixes the crossover between depinning and fast flow.

Figure 4

(a) Dynamical phase diagram for $f<f_c$ in the limit of vanishing temperature. (b) Schematic structure factor for $f<f_c$. The length $L_{\text{opt}}$ fixes the crossover between equilibrium and depinning.

Figure 5

(a) Dynamical phase diagram at finite temperature. (b) Schematic structure factor below $f_c$. The correlation length $\xi$ fixes the crossover between depinning and fast flow, and the length $L_{\text{opt}}$ the crossover between equilibrium and depinning.

Figure 6

Sequence of metastable configurations $\left\{\dots ,{\alpha }_k,{\alpha }_{k+1},\dots \right\}$ detected by our algorithm. For clarity, ${\alpha }_{k+1}$ is advanced by a small amount with respect to ${\alpha }_k$.

Figure 7

Escape path from a metastable configuration ${\alpha }_k$ via a configuration $\beta$ that relaxes toward a new metastable configuration ${\alpha }_{k+1}$ of lower energy.

Figure 8

Ground-state, critical and dominant configurations below $f_c$. An excitation of size $L_{\text{opt}}$ escapes the system from the dominant configuration $\alpha$. The configuration $\beta$ relaxes to the metastable state $\gamma$ via an avalanche of size $L_{\text{relax}}$.

Figure 9

Rescaled structure factor of the harmonic string in the Arrhenius limit (averaged over 1000 disorder realizations) for $L=16,32,64,128$.

Figure 10

Steady-state structure factor for a string with hard metric constraint in the Arrhenius limit (averaged over $10000$ disorder realizations) for $L=M=64$. Curves for different forces are shifted for clarity.

Figure 11

Steady-state structure factor for a tilted string with hard metric constraint in the Arrhenius limit (averaged over 1000 disorder realizations) for $L=M=512,256$. Curves for different forces are shifted for clarity.

Figure 12

Mean escape barrier $U$ from the dominant configuration for samples $L\times L^{{\zeta }_{\text{dep}}}$ with $L=32,64,128$ (the line is a guide to the eye).

Figure 13

Distribution of $U$ for $11000$ disorder realizations (samples $L\times L^{{\zeta }_{\text{dep}}}$ with $L=8,16,32$). The width of the distribution decreases with increasing $L$.

Figure 14

Finite-temperature steady-state structure factor from Langevin simulations of an harmonic string of size $\left(L,M\right)=\left(1024,2048\right)$. The inset shows $S\left(q\right)$ for $T=0.05$ and $f/f_c=0.8,0.85,\dots 1.2$. The crossover length $\xi$ is a fit parameter for the collapse in the main panel.

Figure 15

$\xi$ as a function of $f$ for a string of size $L=1024$ and for different $T$ increasing from the top to the bottom $\left(T=0.0025,0.005,0.01,0.025,0.05,0.075,0.1,0.15,0.2\right)$.

Figure 16

Behavior of the dynamical length $L_{\text{relax}}$. This length diverges with the exponent ${\nu }_{\text{dep}}$ (see inset). The string sizes are $L=16,32,64,128$ for $f/f_c=\left(0.2,0.25\right)$,(0.3–0.4), (0.5–0.75) (0.8–0.9), respectively.

Figure 17

Metastable states of the low-temperature in a $\left(L,M=32,64\right)$ system, for different forces $f<f_c$. The dominant configuration is emphasized.

Figure 18

Decomposition of a configuration $\alpha \left(t\right)$ as a sum of ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$, the configuration $\varphi$ by the sum of ${\varphi }_1$ and ${\varphi }_2$, the configuration ${\alpha }_{\varphi }\left(t\right)$ by the sum of ${\varphi }_1$ and ${\alpha }_2\left(t\right)$, and the configuration ${\varphi }_{\alpha }\left(t\right)$ by the sum of ${\alpha }_1\left(t\right)$ and ${\varphi }_2$.