Creep and thermal rounding close to the elastic depinning threshold

We study the slow stochastic dynamics near the depinning threshold of an elastic interface in a random medium by solving a particularly suited model of hopping interacting particles that belongs to the quenched-Edwards-Wilkinson depinning universality class. The model allows us to compare the cases of uniformly activated and Arrhenius activated hops. In the former case, the velocity accurately follows a standard scaling law of the force and noise intensity with the analog of the thermal rounding exponent satisfying a modified “hyperscaling” relation. For the Arrhenius activation, we find, both numerically and analytically, that the standard scaling form fails for any value of the thermal rounding exponent. We propose an alternative scaling incorporating logarithmic corrections that appropriately fits the numerical results. We argue that this anomalous scaling is related to the strong correlation between activated hops that, alternated with deterministic depinning-like avalanches, occur below the depinning threshold. We rationalize the spatiotemporal patterns by making an analogy of the present model in the near-threshold creep regime with some well-known models with extremal dynamics, particularly the Bak-Sneppen model.

Figure 1

(a) Velocity as a function of the applied force for a qEW model with $3\times {10}^4$ sites, using the narrow well form of the pinning potential. Continuous line is a fitting using a function $v\sim {(f-f_c)}^{\beta }$. The fitted values of $f_c$ and $\beta$ are $f_c=1.600\pm 0.001, \beta =0.24\pm 0.01$. (b) Same data in logarithmic scale, with $\mathrm{\Delta }\equiv (f-f_c)$.

Figure 2

Time-dependent structure factor $S_q\left(t\right)\equiv \langle |u_q\left(t\right){|}^2\rangle$ for an initially flat interface at $t=0$ right at $f_c$, for the model of Eq. (4) with narrow pinning wells. Inset: raw data for different times. Main: scaled data using the universal nonstationary relaxation form $S_q\left(t\right)\sim q^{-(1+2\zeta )}G\left(q{t}^{1/z}\right)$ and the precisely known [31] dynamical and roughness depinning exponents, $z$ and $\zeta$, respectively.

Figure 3

Velocity $v$ as a function of the intensity of the external field $h$, at the critical point. Points are the results of the simulation. Straight red line is drawn with the expected slope ${\psi }_h\simeq 0.082$. The numerical data approach the theoretical prediction for sufficiently low values of $h$.

Figure 4

(a) Velocity as a function of external field for qEW model, below and above the critical point, for different values of $h$, as indicated. The $h=0$ critical behavior is indicated by the dotted line. (b) Same results, rescaled using the expected values of $\beta =0.245$ and ${\psi }_h=0.082$ ($\mathrm{\Delta }=f-f_c$). The continuous line is the asymptotic form predicted by Eq. (7).

Figure 5

Space and time distribution of active sites in the system, for the qEW case. Small (red) dots are sites that are activated by neighbor sites in the previous time step. Larger (black) dots are sites activated at the uniform rate $h$, and thus randomly distributed. The plot was obtained below criticality, at $f=0.54, h={10}^{-3}$. Note the structure of independent clusters, each of them initiated by a site activated by the field. The space-time span of the graph is 1000 sites and 1500 time steps.

Figure 6

Same as previous figure, at $f=f_c\simeq 1.600, h={10}^{-3}$. The span of the graph is 300 sites and 450 time steps.

Figure 7

Energy barrier as a function of the applied elastic force $f_{\text{el}}$ for different forms of the pinning wells. (a) Smooth pinning case. (b) Sharp-ending parabolic pinning well. (c) Triangular well. The energy barrier $\epsilon$ behaves as $\epsilon \sim {(f_c-f_{\text{el}})}^{\alpha }$, with $\alpha =3/2, \alpha =2$, and $\alpha =1$ respectively. Note that for one particle $f_c\equiv f_i^{\text{th}}$.

Figure 8

Same as Fig. 6 but for a finite temperature ($\alpha =1$), instead of an uniform activation rate. Parameters of the simulation are $f=0.5, T=0.01$. Note how now the location of thermally activated sites is strongly correlated spatially and temporally. The span of the graph is 1500 sites and 1000 time steps.

Figure 9

Velocity as a function of applied force, at different temperatures in a system of ${10}^4$ sites with $\alpha =1$. Temperature is $T=0.02$ for the left-most curve, and is divided by two for each successive curve linear (a) and logarithmic (b) scale. In (b), the red straight lines display the form $v=C{T}^{q}exp(\mathrm{\Delta }/T)$, with $q=1.7$ and a single $C$ factor for all curves. For a justification of the $T^q$ factor see Sec. 4b.

Figure 10

Velocity $v$ as a function of temperature at the critical point. The red straight line has a slope $\beta \equiv 0.245$, which is the value predicted by the arguments of Middleton [20]. The numerical results seem to point to a value $\simeq 0.18$, instead (dotted line).

Figure 11

The data in Fig. 9 scaled according to (a) the Middleton suggestion, and (b) using the value 0.18 for the thermal rounding exponent, according to results in Fig. 10. None of the cases permits a correct scaling of the whole data.

Figure 12

From bottom to top: sequence of metastable configurations visited by an elastic string of size $L=128$ driven upwards by the subthreshold forces $f_2=1.0$ (red lines) and $f_1=1.5$ (black circles), in the low-temperature limit for the same realization of the disorder. The initial configuration, at the bottom, is the same for both sequences and is an arbitrary metastable state prepared at $f_1<f_c\approx 1.6$. We find that, in general, any metastable state of the $f_1$-sequence is contained in the $f_2$-sequence, provided that $f_2<f_1$. It follows that the critical configuration itself belongs to any $f$-sequence, provided $f<f_c$.

Figure 13

Distribution of values of $x_i$ in the system after a long equilibration time, using the thermal activation protocol, in the presence of an external force $f$. Note the appearance in most of the system of a “gap” between $f$ and $f_c$. Sites around $i\sim 5500$ have lower values of $x_i$ because of recent avalanches in this region. It is clear that next avalanches will occur around this region too, i.e., highly correlated spatially with previous avalanches.

Figure 14

Sequential index $n$ against the spatial extent of the avalanche. Note the strong spatial correlation between consecutive avalanches.

Figure 15

Left: The value of $x_{\text{min}}$ for the sequence of avalanches in the previous figure. From these values of $x_{\text{min}}$, the values of time in the next figure are obtained as indicated in the text. Dashed line is a reference value used to define clusters. With this threshold, the arrows point to the first and last avalanche that form a particular cluster. Right: histogram of the observed values of $x_{\text{min}}$. The expected exponent for the power law is $2\nu -1$, and this coincides with the observed value.

Figure 16

Time of the $n\mathrm{th}$ avalanche $t_n$ (normalized to the maximum value $t_{\text{max}}$) against spatial extent of the avalanches, obtained for $T=0.002$, and using $\alpha =1$. The largest values of $x_{\text{min}}$ in the previous figure are responsible of the largest temporal intervals between consecutive avalanches in this figure.

Figure 17

Spatial correlations around the depinning transition. The steady-state disorder averaged structure factor $S_q$ at $T=0$ (a) and $T=0.005$ (b) displays a roughness crossover, between $S_q\sim 1/q^{1+2\zeta }$ with $\zeta \approx 1.25$ and $S_q\sim 1/q^2$ at the characteristic length-scale $\xi (f,T)$. (c) Force dependence of $\xi (f,T)$ for the two temperatures. The dashed-line indicates the expected divergent behavior of the $T=0$ depinning transition, $\xi (f,T=0)\sim {(f-f_c)}^{-\nu }$, with $\nu =1/(2-\zeta )\approx 1.33$. At finite temperature the divergence of $\xi$ at $f_c$ disappears, and $\xi (f,T)$ monotonically grows with decreasing $f$ in the creep regime. In either case, the large-scale geometry is described by the critical roughness $\zeta$.

Figure 18

Generalized scaling of the $v(\mathrm{\Delta },T)$ data. Here, the value $T_0=0.1$ is used, but note anyway that any value of $T_0$ would be equally effective.

Figure 19

Symbols reproduce the same data as in Fig. 10. Red line is $\sim T^{\beta }$ and blue one is the expected asymptotic form $\sim {[-Tln(T/T_0)]}^{\beta }$, for $T_0=0.1$. Green line is the result expected using the full form of the function $g\left(x\right)$.

Figure 20

Mean velocity as a function of the driving force and temperature (temperature increases from the bottom to the top curves) for a particle in a periodic potential described by Eq. (A4) with $\gamma =2$ (a) and $\gamma =3$ (b). Both family of curves can be collapsed by rescaling using Eq. (3) with the appropriate critical exponents of Eq. (A3): $\gamma =2$ (c) and $\gamma =3$ (d).