L-5

Goal: This is the first lecture on the dynamics of disordered systems. We will explore how disorder in various systems induces pinning up to a critical threshold. Near this threshold, the dynamics become intermittent and are dominated by large reorganizations known as avalanches.

Pinning and Depinning of a Disordered Material

In earlier lectures, we discussed how disordered systems can become trapped in deep energy states, forming a glass. Today, we will examine how such systems can also be pinned and resist external deformation. This behavior arises because disorder creates a complex energy landscape with numerous features, including minima (of varying depth), maxima, and saddle points.

When an external force is applied, it tilts this multidimensional energy landscape in a specific direction. However, local minima remain stable until a finite critical threshold is reached. Two important dynamical phase transitions are induced by pinning

• The depinning transition: Interfaces pinned by impurities are ubiquitous and range from magnetic domain walls to crack fronts, from dislocations in crystals to vortices in superconductors. Above a critical force, interfaces depin, their motion becomes intermittent, and a Barkhausen noise is detected.

• The yielding transition: Everyday amorphous materials such as mayonnaise, toothpaste, or foams exhibit behavior intermediate between solid and liquid. They deform under small stress (like a solid) and flow under large stress (like a liquid). In between, we observe intermittent plastic events.

Depinning tranition: the equation of motion

In the following we focus on the depinning trasition. At zero temperature and in the overdamped regime, where $\rho \partial _{t}^{2}+{\frac {1}{\mu }}\partial _{t}\approx {\frac {1}{\mu }}\partial _{t}$, the equation of motion for the interface is:

$\partial _{t}h(x,t)=\nabla ^{2}h+f+F(x,h(x,t)),\quad {\text{with}}\;F(x,h(x,t))=-{\frac {\delta V(x,h(x,t))}{\delta h(x,t)}}$

Here we set$\mu =1$, $f$ the external force and the disorder force is $F(x,h(x,t))$. Again we can consider a gaussian force of zero mean and correlations:

${\overline {F(x,h)F(x',h')}}=\delta ^{d}(x-x')\Delta (h-h')$

There are usually two kinds of disorder:

• (i) Random Bond (RB): If $V(x,h)$ is short-range correlated, the area below $\Delta (h)$ is zero.
• (ii) Random Field: If $V(x,h)$ behaves like a Brownian motion along $h$, then $\Delta (h)$ is short-range correlated.

The No-Passing Rule

Interfaces obey the so-called no-passing rule. Consider two interfaces $h_{\alpha }(x,t)$ and $h_{\beta }(x,t)$ such that $h_{\alpha }(x,t=0)<h_{\beta }(x,t=0)$ for every $x$. In the overdamped case, $\alpha$ will never overtake $\beta$.

To see why, assume for contradiction that at some time $t^{*}$, $\alpha$ reaches $\beta$ at a point $x^{*}$, i.e., $h_{\alpha }(x^{*},t^{*})=h_{\beta }(x^{*},t^{*})$. At this point, it can be shown that the local velocity of $\alpha$, denoted by $v_{\alpha }(x^{*},t^{*})$, is strictly less than the local velocity of $\beta$, $v_{\beta }(x^{*},t^{*})$.

This contradiction implies that the no-passing rule holds: $\alpha$ cannot overtake $\beta$. An important consequence of the no-passing rule is that the value of the critical force $f_{c}$ is independent of the initial condition. Indeed, if at a given force $f$ the configuration $\beta$ is a dynamically stable state, it will act as an impenetrable boundary for all configurations preceding it.

When $f=f_{c}$, the system possesses a single dynamically stable configuration. For $f>f_{c}$, no metastable states exist, and the system transitions into a fully moving phase.

Critical Properties at Depinning

• At the critical force $f_{c}$, the interface becomes rough with a new roughness exponent:

$h(bx)\sim b^{\zeta }h(x).$

• Starting from a flat configuration and setting $f=f_{c}$, the roughening process occurs over a growing length scale:

$\ell (t)\sim t^{1/z},$

where $z$ is the dynamical exponent.

At the depinning transition, the interface is not only self-affine but also exhibits critical dynamics characterized by avalanches:

• The order parameter of the transition is the center-of-mass velocity of the interface, which vanishes near the critical force as:

$v_{CM}\sim |f-f_{c}|^{\beta }.$

• The correlation length can be determined from the two-point velocity correlations:

${\overline {v(y+x,t)v(y,t)}}\sim {\frac {e^{-x/\xi (f)}}{|x|^{\kappa }}},\quad \xi (f)\sim |f-f_{c}|^{-\nu },$

where $\xi (f)$ diverges at $f_{c}$ with the exponent $\nu$.

• Below $f_{c}$, the motion of the interface is intermittent, characterized by avalanches. The size distribution of avalanches is scale-free up to a cut-off:

$P(S)=S^{-\tau }f(S/S_{\max }),\quad S_{\max }\sim \xi (f)^{d+\zeta }\sim |f-f_{c}|^{-(d+\zeta )\nu },$

where $d$ is the spatial dimension, and $\tau$ characterizes the avalanche size distribution.

Depinning exponents

Exponent	Observable	Mean field	d=1
$z$	$\ell (t)\sim t^{z}$	2	$1.43\pm 0.01$
$\zeta$	$h(bx)\sim b^{\zeta }h(x)$	0	$1.25\pm 0.01$
$\nu$	$\xi (f)\sim |f-f_{c}|^{-\nu }$	1/2	$\nu ={\frac {1}{2-\zeta }}$
$\beta$	$v_{CM}\sim |f-f_{c}|^{\beta }$	1	$\beta =\nu (z-\zeta )$
$\tau$	$P(S)\sim S^{-\tau }$	3/2	$\tau =2-{\frac {2}{d+\zeta }}$

Cellular Automata

Our goal is to study the avalanches observed as the critical force is approached from below. To this end, we introduce a discrete version of the interface's equation of motion. These cellular automata belong to the same universality class as the original model. They are simple to implement numerically, allow a straightforward definition of avalanches, and can be solved in the mean-field limit. For simplicity, we first discuss the $d=1$ case.

The first step is the disretization along the $x$ direction. The line is now a collection of $L$ blocks connected by springs. The spring constant is set equal to one. Hence the velocity of each block is

$v_{i}(t)=\partial _{t}h_{i}(t)={\frac {1}{2}}(h_{i+1}(t)+h_{i-1}(t)-2h_{i}(t))+f+F_{i}(h_{i}(t)),$

To study avalanches below threshold, one as to trigger them at constant force, which is not very convenient. It is useful to replace the external force with a parabolic potential ${\frac {m^{2}}{2}}(w-h_{i}(t))^{2}$, here $m^{2}$ is the spring constant, try to bring each block at the equilibrium position $w$. The velocity of each block becomes

$v_{i}(t)=\partial _{t}h_{i}(t)=h_{i+1}(t)+h_{i-1}(t)-2h_{i}(t)+m^{2}(w-h_{i}(t))+F_{i}(h_{i}(t)),$

Note that in a stable configuration, where all the velocity are zero, the force acting on the line per unit length is $m^{2}(w-h_{CM}(t))$. Increasing $w$ the force slowly increases if $h_{CM}$ doest not move. When an avalanche occurs the force decreses.

Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along $h_{i}$ impurities act as pinning center that trap the block around their position until a local threshold $\sigma _{i}^{th}$ is reached. In this limit the local velocities are zero when the block is trapped and quickly move to the next impurity once the threshold is overcome. The obteained cellular automata can be described by the following algorithm:

• Drive: Increasing $w$ each point of the interface is pulled with a slowly increasing force or stress:

$\sigma _{i}={\frac {1}{2}}(h_{i+1}+h_{i-1}-2h_{i})+m^{2}(w-h_{i})$

.

• Instability: An instability occurs when $\sigma _{i}$ reaches its $\sigma _{i}^{th}$. Hence the point moves to the next well

${\begin{cases}\sigma _{i}=\sigma _{i}-\Delta \quad {\text{stress drop}}\\\\\sigma _{i\pm 1}=\sigma _{i\pm 1}+{\frac {1}{2}}{\frac {\Delta }{1+m^{2}}}\quad {\text{stress redistribution}}\\\end{cases}}$

Note that $\Delta$ is a positive random variable drwan from $g(\Delta )$.

• Avalanche: The two neighbours can be unstable... An avalanche can start...