Driven Interfaces: From Flow to Creep Through Model Reduction

Abstract

The response of spatially extended systems to a force leading their steady state out of equilibrium is strongly affected by the presence of disorder. We focus on the mean velocity induced by a constant force applied on one-dimensional interfaces. In the absence of disorder, the velocity is linear in the force. In the presence of disorder, it is widely admitted, as well as experimentally and numerically verified, that the velocity presents a stretched exponential dependence in the force (the so-called ‘creep law’), which is out of reach of linear response, or more generically of direct perturbative expansions at small force. In dimension one, there is no exact analytical derivation of such a law, even from a theoretical physical point of view. We propose an effective model with two degrees of freedom, constructed from the full spatially extended model, that captures many aspects of the creep phenomenology. It provides a justification of the creep law form of the velocity–force characteristics, in a quasistatic approximation. It allows, moreover, to capture the non-trivial effects of short-range correlations in the disorder, which govern the low-temperature asymptotics. It enables us to establish a phase diagram where the creep law manifests itself in the vicinity of the origin in the force–system-size–temperature coordinates. Conjointly, we characterise the crossover between the creep regime and a linear-response regime that arises due to finite system size.

Appendices

Appendix 1: Statistical Tilt Symmetry at $f\ne 0$

We derive in this appendix a form of the STS, which allows to decompose the point-to-point free energy into the sum of a deterministic contribution and of disorder-dependent one, which, in distribution, is invariant by translation along direction y. We first recall the known result at zero force before deriving a novel STS at $f\ne 0.$

1.1 A Reminder: The Zero Force STS ($f=0$)

In the zero force situation, the linear change of coordinates $y(t)= \tilde{y}(t) + y_{\text {i}}+ \frac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}} t$ allows to relate the weight of trajectories starting in $(0,\,y_{\text {i}})$ and arriving in $(t_{\text {f}},\,y_{\text {f}})$ (see Fig.2) to those starting in (0, 0) and arriving in $(t_{\text {f}},\,0)$ in a translated disorder. One directly reads from (15) (with $f=0$) that

$$\begin{aligned} W_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) = \text {e}^{-\frac{c}{T} \frac{(y_{\text {f}}-y_{\text {i}})^2}{2 t_{\text {f}}}} W_{\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}}} V}\left( t_{\text {f}},\,0|0,\,0\right) , \end{aligned}$$

(A.77)

where the translated disorder $\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}}} V$ is defined as

$$\begin{aligned} \mathcal T_{y_{\text {f,i}}}^{t_{\text {f}}} V(t,\,\tilde{y})\equiv V\big (t,\,\tilde{y}+y_{\text {i}}+\tfrac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}}t\big ). \end{aligned}$$

(A.78)

This remark enables to decompose the free energy $F_V(t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}})={-}T\log W_V(t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}})$ as

$$\begin{aligned} F_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) = F_{V\equiv 0}\left( t_{\text {f}},\,y_{\text {f}}-y_{\text {i}}\right) + \bar{F}_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) , \end{aligned}$$

(A.79)

where

$$\begin{aligned} F_{V\equiv 0}\left( t_{\text {f}},\,y_{\text {f}}-y_{\text {i}}\right)&= F_{\text {th}}\left( t_{\text {f}},\,y_{\text {f}}-y_{\text {i}}\right) + \frac{T}{2} \log \frac{2\pi Tt}{c}, \end{aligned}$$

(A.80)

$$\begin{aligned} F_{\text {th}}\left( t_{\text {f}},\,y_{\text {f}}-y_{\text {i}}\right)&= c \frac{(y_{\text {f}}-y_{\text {i}})^2}{2 t_{\text {f}}}. \end{aligned}$$

(A.81)

Hence, $\bar{F}_V(t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}})$ is invariant by translation in distribution, as we indeed observe that it is depending on $y_{\text {i}}$ and $y_{\text {f}}$ only through $\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}}}V.$

$$\begin{aligned} \bar{F}_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) = {-}T \log W_{\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}}} V}\left( t_{\text {f}},\,0|0,\,0\right) . \end{aligned}$$

(A.82)

1.2 The Non-zero Force STS ($f\ne 0$)

In the non-zero force situation, the key observation is that there exists a f-dependent transformation of the directed polymer trajectories

$$\begin{aligned} y(t)= \tilde{y}(t) + y_{\text {i}}+ \frac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}} t + \frac{f}{2c}t\left( t_{\text {f}}-t\right) , \end{aligned}$$

(A.83)

which, remarkably, implies a f-STS generalising (A.79):

$$\begin{aligned} W^f_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) = \exp \Big \{ -\frac{1}{T} \Big [ c \frac{(y_{\text {f}}-y_{\text {i}})^2}{2 t_{\text {f}}}-\frac{f}{2} t_{\text {f}}\left( y_{\text {f}}+y_{\text {i}}\right) -\frac{f^2}{24 c}t_{\text {f}}^3 \Big ] \Big \} W_{\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}},f} V}\left( t_{\text {f}},\,0|0,\,0\right) . \end{aligned}$$

(A.84)

The result is obtained by coming back to the path-integral definition of the f-dependent point-to-point partition function (15), and taking care of the boundary terms. The key observation in the computation is to recognise the following total derivative

$$\begin{aligned} \frac{ (2 c (y_{\text {f}}-y_{\text {i}})+f t_{\text {f}}(t_{\text {f}}-2 t))}{2 t_{\text {f}}}\partial _t \tilde{y}(t)-f \tilde{y}(t) = \partial _t\Big [ \frac{f t_{\text {f}}( t_{\text {f}}- 2 t) + 2 c (y_{\text {f}}- y_{\text {i}}) }{2 t_{\text {f}}} \tilde{y}(t) \Big ], \end{aligned}$$

(A.85)

for the terms linear in $\tilde{y}(t)$ after the change of variable (A.84). The translated disorder $\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}},f} V^{\!f}\!$ is defined as

$$\begin{aligned} \mathcal T_{y_{\text {f,i}}}^{t_{\text {f}},f} V(t,\,\tilde{y})\equiv V\big (t,\,\tilde{y}+y_{\text {i}}+\tfrac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}}t+ \tfrac{f}{2c}t\left( t_{\text {f}}-t\right) \big ). \end{aligned}$$

(A.86)

Note importantly that if the translated disorder (A.86) depends on the force through a f-dependent change of coordinate, the partition function on the right hand side of (A.84) is however the one at zero force. This enables to decompose the free energy $F_V^f(t,\,y)$ as

$$\begin{aligned} F_V^f\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right)= & {} F_{\text {th}}\left( t_{\text {f}},\,y_{\text {f}}-y_{\text {i}}\right) -\frac{ft_{\text {f}}}{2} \left( y_{\text {f}}+y_{\text {i}}\right) \nonumber \\&-\,\frac{f^2}{24 c}t_{\text {f}}^3 + \bar{F}^f_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) + \frac{T}{2} \log \frac{2\pi Tt}{c}, \end{aligned}$$

(A.87)

where

$$\begin{aligned} \bar{F}^f_V\left( t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}}\right) = {-}T \log W_{\mathcal T_{y_{\text {f,i}}}^{t_{\text {f}},f} V}\left( t_{\text {f}},\,0|0,\,0\right) . \end{aligned}$$

(A.88)

For uncorrelated disorder ($\xi =0$) this implies in particular that at large $t_{\text {f}}$ the distribution of $\bar{F}^f_V(t_{\text {f}},\,y_{\text {f}}|0,\,y_{\text {i}})$ adopts the Airy$_2$ scaling [10] and goes to the distribution of a Brownian motion of coordinate $y_{\text {f}}-y_{\text {i}}$ at infinite $t_{\text {f}}$ (this corresponds to the steady state of the KPZ equation [19]). For correlated disorder ($\xi >0$), the picture is slightly changed at large $t_{\text {f}}{\text {:}}$ the steady state is not distributed as a Brownian anymore but scales similarly (as long as $y_{\text {f}}-y_{\text {i}}\gg \xi$) with a different $\xi$-dependent amplitude $\tilde{D}$ [11, 52] [see Eq. (42) in the main text].

Appendix 2: Effective Temperature $\tilde{T}$ and Friction $\tilde{\gamma }$ in the Absence of Disorder

In this Appendix, we determine the relation between the original friction $\gamma$ and temperature T of the interface dynamics (1) and the ones $\tilde{\gamma }$ and $\tilde{T}$ of the effective model (33–34), in the absence of disorder ($V\equiv 0$). Starting by the original dynamics, one averages (1) over thermal fluctuations and taking the long-time limit, one gets

$$\begin{aligned} \overline{v}(f)\big |_{V\equiv 0} = \frac{f}{\gamma }. \end{aligned}$$

(B.89)

Changing to the reference frame of the centre of mass, one recognises that $y(t,\,\tau )-\frac{f}{\gamma }\tau$ obeys the Edwards–Wilkinson equation, whose steady state is Gaussian, implying that

$$\begin{aligned} \big \langle \left[ y\left( t_{\text {f}}\right) -y(0)\right] ^2\big \rangle \big |_{V\equiv 0} = \frac{T}{c}t_{\text {f}}. \end{aligned}$$

(B.90)

On the other hand, the effective equations (33–34) are written as follows in the absence of disorder, as seen from the expression (A.87) of the tilted free energy:

$$\begin{aligned} \tilde{\gamma } \partial _\tau y_{\text {i}}(\tau )&= {+}\,c\frac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}}+\frac{1}{2} ft_{\text {f}}+ \sqrt{2\tilde{\gamma }\tilde{T}}\tilde{\eta }_\text {i}(\tau ), \end{aligned}$$

(B.91)

$$\begin{aligned} \tilde{\gamma } \partial _\tau y_{\text {f}}(\tau )&= {-}c\frac{y_{\text {f}}-y_{\text {i}}}{t_{\text {f}}}+\frac{1}{2} ft_{\text {f}}+ \sqrt{2\tilde{\gamma }\tilde{T}}\tilde{\eta }_\text {f}(\tau ). \end{aligned}$$

(B.92)

Summing these equations, averaging over thermal noise and imposing that both $y_{\text {i}}(\tau )$ and $y_{\text {i}}(\tau )$ move at an average velocity $\overline{v}(f)|_{V\equiv 0}$ at large times, one gets $2\tilde{\gamma }\overline{v}(f)|_{V\equiv 0} = ft_{\text {f}};$ hence, comparing to (B.89), one obtains

$$\begin{aligned} \tilde{\gamma } = \frac{1}{2} t_{\text {f}}\gamma . \end{aligned}$$

(B.93)

Note that to get this result, we relaxed the conditions (walls) that would enforce the model to reach a zero-velocity equilibrium steady state at large times. However, a MFPT analysis in such conditions would also yield the result (B.93) for the velocity in a finite window of the system far from the walls. Subtracting now (B.91) and (B.92) one obtains a closed equation for $\delta \!y(\tau )=y_{\text {f}}(\tau )-y_{\text {i}}(\tau )$ in the absence of disorder:

$$\begin{aligned} \tilde{\gamma }\partial _\tau \delta \!y(\tau ) = {-}2\frac{c}{t_{\text {f}}}\delta \!y(\tau ) + \sqrt{4\tilde{\gamma }\tilde{T}}\tilde{\eta }_1, \end{aligned}$$

(B.94)

where $\tilde{\eta }_1$ is a white noise of unit variance. The force term of this Langevin equation derives from the energy $\frac{c}{t_{\text {f}}}(\delta \!y)^2$ and the noise term has temperature $2\tilde{T}.$ This shows that the steady-state distribution of $\delta \!y$ is Gaussian $\propto \exp \left[ -\frac{c}{2\tilde{T} t_{\text {f}}}(\delta \!y)^2\right] .$ This implies in turn that $\langle (\delta \!y)^2\rangle |_{V\equiv 0}=\langle [y_{\text {f}}-y_{\text {i}}]^2\rangle |_{V\equiv 0}=\frac{\tilde{T}}{c}t_{\text {f}}.$ Finally, comparing to (B.90) one obtains that the temperatures of the original and of the effective models are equal:

$$\begin{aligned} \tilde{T} = T. \end{aligned}$$

(B.95)

To summarise, in the absence of disorder, the effective model with friction (B.93) and temperature (B.95) presents the same velocity and the same Gaussian end-point distribution as the original dynamics. We assume that this correspondence between original and effective parameters also holds in the presence of disorder.