LBan-V

Avalanches at the Depinning Transition

In the previous lesson, we studied the dynamics of an interface in a disordered medium under a uniform external force $F$. In a fully connected model, we derived the force–velocity characteristic and identified the critical depinning force $F_c$.

In this lesson, we focus on the avalanches that occur precisely at the depinning transition. To do so, we introduce a new driving protocol: instead of controlling the external force $F$, we control the position of the interface by coupling it to a parabolic potential. Each block is attracted toward a prescribed position $w$ through a spring of stiffness $k_0$.

For simplicity, we restrict to the fully connected model, where the distance of block $i$ from its local instability threshold is

$x_i=1-(h_{CM}-h_i)-k_0(w-h_i).$

The first term represents the elastic pull exerted by the rest of the interface, while the second encodes the external driving through the spring.

Here $h_{CM}$ is the center-of-mass position of the interface. Because the sum of all internal elastic forces vanishes, the total external force is balanced solely by the pinning forces. The effective external force can thus be written as a function of $w$:

$F(w)=k_0(w-h_{CM}).$

As $w$ is increased quasistatically, the force $F(w)$ would increase if $h_{CM}$ were fixed. When an avalanche takes place, $h_{CM}$ jumps forward and $F(w)$ suddenly decreases. However, in the steady state and in the thermodynamic limit $L\to \mathrm{\infty }$, the force recovers a well-defined value. In the limit $k_0\to 0$, this force tends to the critical depinning force $F_c$; at finite $k_0$ it lies slightly below $F_c$.

Quasi-Static Protocol and Avalanche Definition

To study avalanches, the position $w$ is increased quasi-statically: it is shifted by an infinitesimal amount $w\to w+\mathrm{d}w$ so that the block closest to its instability threshold reaches it,

$x_i=0.$

This block is the epicenter of the avalanche: it becomes unstable and jumps to the next well.

When block $i$ jumps by $\mathrm{\Delta }$, both the elastic contribution and the driving spring relax. This gives

$\{\begin{array}{l}{x}_i=0⟶x_i=\mathrm{\Delta }(1+k_0),\\ x_j⟶x_j-\frac{\mathrm{\Delta }}{L}(j\ne i).\end{array}$

The key feature of the quasi-static protocol is that $w$ does not evolve during the avalanche: all subsequent destabilizations are triggered exclusively by previously unstable blocks.

It is convenient to organize the avalanche into generations of unstable sites:

• First generation: the epicenter.
• Second generation: sites destabilized by it.
• Third generation: sites destabilized by generation two.
• And so on.

This hierarchical construction allows us to compute avalanche amplification step by step.

Derivation of the Evolution Equation

Our goal is to determine the distribution $P_w(x)$ of distances to threshold at fixed $w$.

We shift the parabola by $w\to w+\mathrm{d}w$. Before the shift:

$x_i(w)=1-k_0(w-h_i(w))+(h_{CM}(w)-h_i(w)).$

We now follow the dynamics generation by generation.

First generation

During the shift, the center of mass has not yet moved.

• Stable sites ($x_i>k_{0}dw$):

$x_i^{t=1}=x_i-k_{0}dw.$

• Sites with $0<x_i<k_{0}dw$ become unstable.

Since $dw$ is infinitesimal, their fraction is

$P_w(0)k_{0}dw.$

They jump and stabilize at

$x_i^{t=1}=\mathrm{\Delta }(1+k_0).$

Second generation

The parabola is now fixed, but the center of mass has advanced:

$h_{CM}\to h_{CM}+\bar{\mathrm{\Delta }}{P}_w(0)k_{0}dw.$

Thus all sites shift again toward instability.

• Stable sites:

$x_i^{t=2}=x_i-(1+\bar{\mathrm{\Delta }}{P}_w(0))k_{0}dw.$

• Newly unstable fraction:

$(1+\bar{\mathrm{\Delta }}{P}_w(0))P_w(0)k_{0}dw.$

Higher generations

Iterating produces a geometric amplification:

$1+\bar{\mathrm{\Delta }}{P}_w(0)+(\bar{\mathrm{\Delta }}{P}_w(0){)}^2+\dots =\frac{1}{1-\bar{\mathrm{\Delta }}{P}_w(0)}.$

The quantity $\bar{\mathrm{\Delta }}{P}_w(0)$ plays the role of a branching ratio: it measures the average number of sites destabilized by one instability.

We obtain

$x\to x-\frac{k_0}{1-\bar{\mathrm{\Delta }}{P}_w(0)}dw.$

and a fraction

$\frac{P_w(0)}{1-\bar{\mathrm{\Delta }}{P}_w(0)}{k}_{0}dw$

is reinjected at $x=\mathrm{\Delta }(1+k_0)$.

This yields

${\partial }_w{P}_w(x)=\frac{k_0}{1-\bar{\mathrm{\Delta }}{P}_w(0)}[{\partial }_x{P}_w(x)+\frac{P_w(0)}{1+k_0}g\left(\frac{x}{1+k_0}\right)].$

Stationary solution

At large $w$:

$0={\partial }_x{P}_{\text{stat}}(x)+\frac{P_{\text{stat}}(0)}{1+k_0}g\left(\frac{x}{1+k_0}\right).$

Solving:

$P_{\text{stat}}(x)=\frac{1}{\bar{\mathrm{\Delta }}(1+k_0)}\underset{x/(1+k_0)}{\overset{\mathrm{\infty }}{\int }}g(z)dz.$

Avalanches

We consider an avalanche starting from a single unstable site $x_0=0$.

Ordering sites by stability:

$x_1<x_2<x_3<\dots$

From order statistics:

$\underset{0}{\overset{x_1}{\int }}{P}_w(t)dt=\frac{1}{L}.$

Thus

$x_n\sim \frac{n}{L{P}_w(0)}.$

Each instability gives kicks $\mathrm{\Delta }/L$.

Compare mean kick and mean gap:

$\frac{\bar{\mathrm{\Delta }}}{L}\text{vs}\frac{1}{L{P}_w(0)}.$

Criticality occurs when

$\bar{\mathrm{\Delta }}{P}_w(0)=1.$

Using the stationary solution:

$\bar{\mathrm{\Delta }}{P}_w(0)=\frac{1}{1+k_0}.$

Hence:

• $k_0>0$ → subcritical.
• $k_0=0$ → critical.

Mapping to Brownian motion

Define

${\eta }_n=\frac{{\mathrm{\Delta }}_n}{L}-(x_n-x_{n-1}).$

The walk

$X_n=\sum _{i=1}^n{\eta }_i$

remains positive while the avalanche propagates.

Avalanche size = first-passage time.

Critical case

Zero drift → Sparre–Andersen theorem:

$P(S)\sim S^{-3/2}.$

Finite $k_0$

Small negative drift → cutoff:

$S_{\max }\sim k_0^{-2}.$

Derivation of the Evolution Equation

Our goal is to determine the distribution $P_w(x)$ of the distances to threshold of all blocks, given their initial distribution $P_0(x)$ and a value of $w$. To derive the evolution equation of $P_w(x)$ we perform an infinitesimal change in the position of the parabolic potential $w\to w+\mathrm{d}w$. The expression of the distance to threshold of block $i$ just before the change is:

$x_i(w)=1-k_0(w-h_i(w))+(h_{CM}(w)-h_i(w)).$

After the change $w\to w+\mathrm{d}w$ , we organize the complex dynamics generation by generation, indexed by a generation time $t=1,2,\dots$:

• At time $t=1$ (first generation):

While the parabola position changes, the center of mass is still $h_{CM}(w)$. Two things can happen:

1. Stable blocks: if $x_i(w)>k_0\mathrm{d}w$, the block approaches its threshold:

$x_i^{t=1}(w+\mathrm{d}w)=x_i(w)-k_0\mathrm{d}w.$

2. Unstable blocks: if $0<x_i(w)<k_0\mathrm{d}w$, the block is unstable and is stabilized. Since $\mathrm{d}w$ is infinitesimal, $x_i(w)\approx 0$. Hence, the fraction of the unstable blocks is $P_w(0)k_0\mathrm{d}w$ and the stabilization is simple:

$x_i^{t=1}(w+\mathrm{d}w)=\mathrm{\Delta }(1+k_0).$

• At time $t=2$ (second generation):

The parabola position remains fixed, but the center of mass advances $h_{CM}^{t=2}(w+\mathrm{d}w)\to h_{CM}(w)+\bar{\mathrm{\Delta }}{P}_w(0)k_0\mathrm{d}w$. Again, two things can happen:

1. Stable blocks: if $x_i(w)>(1+\bar{\mathrm{\Delta }}{P}_w(0))k_0\mathrm{d}w$, the block approaches its threshold:

$x_i^{t=2}(w+\mathrm{d}w)=x_i^{t=1}(w+\mathrm{d}w)-\bar{\mathrm{\Delta }}{P}_w(0)k_0\mathrm{d}w=x_i(w)-(1+\bar{\mathrm{\Delta }}{P}_w(0))k_0\mathrm{d}w.$

2.Unstable blocks: if $x_i(w)<(1+\bar{\mathrm{\Delta }}{P}_w(0))k_0\mathrm{d}w$, the block is unstable and is stabilized:

$x_i^{t=1}(w+\mathrm{d}w)=\mathrm{\Delta }(1+k_0).$

The total fraction of unstable blocks is $(1+\bar{\mathrm{\Delta }}{P}_w(0))P_w(0)k_0\mathrm{d}w$.

• At the end:

This procedure can be iterated to higher generations $t=3,4,\dots$ and is at the origin of a geometric series:

$(1+\bar{\mathrm{\Delta }}{P}_w(0)+(\bar{\mathrm{\Delta }}{P}_w(0){)}^2+\dots )=\frac{1}{1-\bar{\mathrm{\Delta }}{P}_w(0)}$

1. The stable blocks approaches their threshold:

$x_i(w+\mathrm{d}w)=x_i(w)-\frac{k_0}{1-\bar{\mathrm{\Delta }}{P}_w(0)}\mathrm{d}w.$

2. A fraction $\frac{P_w(0)}{1-\bar{\mathrm{\Delta }}{P}_w(0)}{k}_0\mathrm{d}w$ of the blocks is unstable and is stabilized at a value $x=\mathrm{\Delta }(1+k_0)$ with a probability $g(\mathrm{\Delta })d\mathrm{\Delta }=g(\frac{x}{1+k_0})\frac{dx}{1+k_0}$. We can finally write the evolution equation

$P_{w+dw}(x)=P_w(x+\frac{k_{0}dw}{1-\bar{\mathrm{\Delta }}{P}_w(0)})+\frac{g(\frac{x}{1+k_0})}{1+k_0}\frac{P_w(0)}{1-\bar{\mathrm{\Delta }}{P}_w(0)}{k}_0\mathrm{d}w.$

The final evolution equation

${\partial }_w{P}_w(x)=\frac{k_0}{1-\bar{\mathrm{\Delta }}{P}_w(0)}({\partial }_x{P}_w(x)+\frac{P_w(0)}{1+k_0}g(\frac{x}{1+k_0}))$

Stationary solution

Increasing the value $w$, the distribution converge to the fixed point:

$0={\partial }_x{P}_{\text{stat}}(x)+\frac{g(\frac{x}{1+k_0})P_w(0)}{1+k_0}$

• Determine $P_{\text{stat}}(0)=\frac{1}{\bar{\mathrm{\Delta }}(1+k_0)}$ using

$1=\underset{0}{\overset{\mathrm{\infty }}{\int }}dx{P}_{\text{stat}}(x)=-\underset{0}{\overset{\mathrm{\infty }}{\int }}dxx{\partial }_x{P}_{\text{stat}}(x)$

• Show

$P_{\text{stat}}(x)=\frac{1}{\bar{\mathrm{\Delta }}(1+k_0)}\underset{x/(1+k_0)}{\overset{\mathrm{\infty }}{\int }}g(z)dz$

which is well normalized.

Critical Force

The average distance from the threshold gives a simple relation for the force acting on the system, namely $1-F=\bar{x}$. In the limit $k_0\to 0$ for the automata model we obtain:

$F_c=1-\underset{0}{\overset{\mathrm{\infty }}{\int }}dxx{P}_{\text{stat}}(x)=1-\frac{1}{2}\frac{\overline{{\mathrm{\Delta }}^2}}{\bar{\mathrm{\Delta }}}$

Avalanches

We consider an avalanche starting from a single unstable site $x_0=0$ and the sequence of sites more close to instabitity $x_1<x_2<x_3\dots$. For each unstable block, all the blocks receive a random kick:

$\frac{{\mathrm{\Delta }}_1}{L},\frac{{\mathrm{\Delta }}_2}{L},\frac{{\mathrm{\Delta }}_3}{L},\dots$

with ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,\dots$ drwan from $g(\mathrm{\Delta })$ Are these kick able to destabilize other blocks?

Given the initial condition and $w$, the state of the system is described by $P_w(x)$. From the extreme values theory we know the equation setting the average position of the most unstable block is

$\underset{0}{\overset{x_1}{\int }}{P}_w(t)dt=\frac{1}{L}$

Hence, for large systems we have

$x_1\sim \frac{1}{L{P}_w(0)},x_2\sim \frac{2}{L{P}_w(0)},x_3\sim \frac{3}{L{P}_w(0)},\dots$

Hence we need to compare the mean value of the kick with the mean gap between nearest unstable sites:

$\frac{\bar{\mathrm{\Delta }}}{L}\text{versus }\frac{1}{P_w(0)L}$

Note that $L$ simplifies. We expect three possibilities:

• if the mean kick is smaller than the mean gap the system is subcritical and avalanches quickly stops.
• if the mean kick is equal to the mean gap the system is critical and avalanches are power law distributed
• if the mean kick is larger of the mean gap the system is super-critical and avalanches are unstable.

Note that in the stationary regime the ratio between mean kick and mean gap is $1/(1+k_0)$. Hence, the system is subcritical when $k_0>0$ and critical for $k_0=0$

Mapping to the Brownian motion

Let's define the random jumps and the associated random walk

${\eta }_1=\frac{{\mathrm{\Delta }}_1}{L}-x_1,{\eta }_2=\frac{{\mathrm{\Delta }}_2}{L}-(x_2-x_1),{\eta }_3=\frac{{\mathrm{\Delta }}_3}{L}-(x_3-x_2)\dots$ $X_n=\sum _{i=1}^n{\eta }_i\text{with}\overline{{\eta }_i}=\frac{\bar{\mathrm{\Delta }}}{L}-\frac{1}{L{P}_w(0)}$

An avalanche is active until $X_n$ is positive. Hence, the size of the avalanche identifies with first passage time of the random walk.

• Critical case : In this case the jump distribution is symmetric and we can set $X_0=0$. Under these hypothesis the Sparre-Andersen theorem state that the probability that the random walk remains positive for $n$ steps is independent on the jump disribution and for a large number of steps becomes $Q(n)\sim \frac{1}{\sqrt{\pi n}}$. Hence, the distribution avalanche size is

$P(S)=Q(S)-Q(S+1)\sim \frac{1}{\sqrt{\pi S}}-\frac{1}{\sqrt{\pi (S+1)}}\sim \frac{1}{2\sqrt{\pi }}\frac{1}{S^{3/2}}$

This power law is of Gutenberg–Richter type. The universal exponent is $\tau =3/2$

• Stationary regime: Replacing $\frac{1}{L{P}_w(0)}$ with $\frac{1}{L{P}_{\text{stat}}(0)}=\frac{\overline{\mathrm{\Delta }(1+k_0)}}{L}$ we get $\overline{{\eta }_i}\sim -k_0\frac{\bar{\mathrm{\Delta }}}{L}$. For small $k_0$ , the random walk is only sliglty tilted. The avalanche distribution will be power law distributed with $\tau =3/2$ until a cut-off

$S_{\max }\sim k_0^{-2}$