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.

Step 1: Discretization along the x direction

The interface is represented as a collection of blocks $i=1,\ldots ,L$ connected by springs, where the spring constant is set to one. The velocity of the $i$-th block is given by:

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

In this setup, the external force $f$ is the control parameter. For a given $f$ the interface reaches a dynamically stable configuration. An infinitesimal increase of math>f</math> triggers an avalanche. However, to gather avalanche size statistics, this approach requires finding a new dynamically stable configuration at the same $f$ as before, which is computationally inefficient.

Step 2: Displacement control protocol

An alternative is to control the displacement of the center of mass, $h_{\text{CM}}$​, of the dynamically stable configuration. To achieve this, we introduce a parabolic potential which attracts each block to position $w$ with a spring constant $m^{2}$. The velocity of each block becomes

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

In a dynamically stable configuration, the velocity of all blocks is zero. Consequently, the sum of all internal elastic forces is also zero because the interface is at equilibrium. The driving force is balanced only by the pinning forces. Hence the external force is a function of $w$

$f(w)=m^{2}(w-h_{CM}).$

As $w$ increases, the force $f$ increases if $h_{CM}$ does not move. When an avalanche occurs, $f$ decreases. However, in the steady state and in the thermodynamic limit ($L\to \infty$), a well-defined value of $f$ is recovered. In the limit $m\to 0$ this force reaches the critical value $f_{c}$, while at finite $m$ is slightly below.

Step 3: Discretization along the h direction

Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. In this approximation, impurities act as pinning centers along trapping the block of the interface around their position $h_{i}$. This simplifies the analysis of the interface dynamics.

To be more precise, the force pulling the $i$-th block is given by:

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

The force $\sigma _{i}$ is a local force, while the external force per unit length is

$f(w)={\frac {1}{L}}\sum _{i=1}^{L}\sigma _{i}=m^{2}(w-h_{CM})$

The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold $\sigma _{i}^{th}$. This threshold can be random but is set equal to 1 here for simplicity.

As $w$ increases, each point of the interface is pulled with a slowly increasing force. An instability occurs when:

$\sigma _{i}\geq \sigma _{i}^{th}$

At this point, the block moves to the next well, triggering the following events:

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

Here, $\Delta$ is a positive random variable drawn from a distribution $g(\Delta )$, representing the magnitude of the drop in the local force. The distance between the wells is given by $(\Delta -m^{2})/2$. After an instability, one of the two neighbors of the unstable block can become unstable. This marks the beginning of an avalanche.

Once the avalanche is over, its size $S$ is a random variable defined as:

$S={\frac {1}{2}}\sum _{i=1}^{N}(\Delta _{i}-m^{2})$

where $N$ is the number of instabilities, which is itself a random number. Note that by approximating $\sum _{i=1}^{N}\Delta _{i}$ with $N{\overline {\Delta }}$, where ${\overline {\Delta }}$ is the average stress drop, the size of the avalanche becomes proportional to $N$:

$S\propto N$

This simple relationship highlights the link between the number of instabilities and the total avalanche size, providing insight into the scaling behavior of avalanches in this system.