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 local force acting on block $i$ is

$\sigma _{i}=h_{CM}-h_{i}+k_{0}\,(w-h_{i}).$

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 $F$ 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 $L\to \infty$, 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 $F_{c}$.

As in the previous lesson, it is convenient to introduce the variables $x_{i}$ that measure the distance of block $i$ from its local instability threshold:

$x_{i}=1-\sigma _{i}.$

These variables are the natural starting point for describing avalanches and their statistics.

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, i.e.

$x_{i}=0.$

This block is the epicenter of the avalanche: it becomes unstable and jumps to the next well. Its stabilization induces a redistribution of the stress over all other blocks, which may in turn become unstable:

${\begin{cases}x_{i}=0\;\longrightarrow \;x_{i}=\Delta (1+k_{0}),\\[6pt]x_{j}\;\longrightarrow \;x_{j}-{\dfrac {\Delta }{L}}\quad {\text{for }}j\neq i.\end{cases}}$

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:

• The first generation consists of the single epicenter block.
• The second generation is formed by the blocks destabilized by the stabilization of the epicenter.
• The third generation consists of the blocks destabilized by the stabilization of the second-generation blocks, and so on.

This hierarchical picture allows us to characterize the size and temporal structure of avalanches.

Avalanche Size

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

$S={\frac {1}{2}}\sum _{i=1}^{N}{\bigl (}\Delta _{i}-k_{0}{\bigr )}$

where $N$ is the total number of instabilities (itself a random variable) and $\Delta _{i}$ is the jump of block $i$.

Approximating the sum as $\sum _{i=1}^{N}\Delta _{i}\approx N\,{\overline {\Delta }}$ makes explicit the proportionality between the number of instabilities and the total avalanche size:

$S\approx {\frac {N}{2}}{\bigl (}{\overline {\Delta }}-k_{0}{\bigr )}.$

This relationship highlights that larger avalanches correspond to a larger number of destabilized blocks.

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:

$x_{i}(w)=1-k_{0}{\bigl (}w-h_{i}(w){\bigr )}+{\bigl (}h_{CM}(w)-h_{i}(w){\bigr )}.$

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):

The center of mass is still $h_{CM}(w)$ and two things can happen:

1. Stable blocks: if $x_{i}(w)>k_{0}\,\mathrm {d} w$, the block approaches its threshold:

$x_{i}^{t=0}(w+\mathrm {d} w)=x_{i}(w)-k_{0}\,\mathrm {d} w.$

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:

$x_{i}^{t=0}(w+\mathrm {d} w)=\Delta .$

• At time $t=2$ (second generation):

The position of the parabolic potential remains fixed at $w+\mathrm {d} w$, but the center of mass of the interface advances by

${\overline {\Delta }}\,P_{w}(0)\,k_{0}\,\mathrm {d} w.$

Again, two cases are possible:

1. Stable blocks: if $x_{i}^{t=0}(w+\mathrm {d} w)>{\overline {\Delta }}\,P_{w}(0)k_{0}\,\mathrm {d} w$, the block approaches its threshold:

$x_{i}^{t=1}(w+\mathrm {d} w)=x_{i}^{t=0}(w+\mathrm {d} w)-{\overline {\Delta }}\,P_{w}(0)\,k_{0}\,\mathrm {d} w.$

2.Unstable blocks: if $0<x_{i}^{t=0}(w+\mathrm {d} w)<k_{0}\,\mathrm {d} w$, the block becomes unstable and is stabilized. As before, since $\mathrm {d} w$ is infinitesimal and $x_{i}^{t=0}(w+\mathrm {d} w)\approx 0$:

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

This event occurs with probability ${\overline {\Delta }}\,P_{w}(0)\,k_{0}\,\mathrm {d} w$.

This procedure can be iterated to higher generations $t=3,4,\dots$ until the avalanche stops.

Dynamics

Our goal is thus to determine the distribution $P_{w}(x)$ of all blocks, given their intial distribution, $P_{0}(x)$, and a value of $w$. Let's decompose in steps the dynamics

• Drive: Increasing $w\to w+dw$ each block decreases its distance to threshold

$x_{i}\to x_{i}-m^{2}dw$

.

As a consequence

$P_{w+dw}(x)=P_{w}(x+m^{2}dw)\sim P_{w}(x)+m^{2}dw\partial _{x}P_{w}(x)$

• Stabilization : A fraction $m^{2}dwP_{w}(0)$ of the blocks is unstable. The stabilization induces the change $m^{2}dwP_{w}(0)\to m^{2}dwP_{w}(0)g(x)$. Hence, one writes

$\partial _{w}P_{w}(x)\sim m^{2}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]$

The stabilization of the unstable blocks induce a drop of the force per unit length

$m^{2}dwP_{w}(0)\int dxxg(x)=m^{2}dwP_{w}(0){\overline {\Delta }}$

\

• Redistribution This drop is (partially) compensated by the redistribution. The force acting on all points is increased:

$x_{i}\to x_{i}-m^{2}dwP_{w}(0){\frac {\overline {\Delta }}{1+m^{2}}}$

Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write

$\partial _{w}P_{w}(x)=m^{2}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]\left[1+P_{w}(0){\frac {\overline {\Delta }}{1+m^{2}}}+(P_{w}(0){\frac {\overline {\Delta }}{1+m^{2}}})^{2}+\ldots \right]$

and finally:

$\partial _{w}P_{w}(x)={\frac {m^{2}}{1-P_{w}(0){\frac {\overline {\Delta }}{1+m^{2}}}}}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]$

Stationary solution

Increasing the drive the distribution converge to the fixed point:

$0=\partial _{x}P_{\text{stat}}(x)+P_{\text{stat}}(0)g(x)$

• Determine $P_{\text{stat}}(0)={\frac {1}{\overline {\Delta }}}$ using

$1=\int _{0}^{\infty }dx\,P_{\text{stat}}(x)=-\int _{0}^{\infty }dx\,x\partial _{x}P_{\text{stat}}(x)$

• Show

$P_{\text{stat}}(x)={\frac {1}{\overline {\Delta }}}\int _{x}^{\infty }g(z)dz$

which is well normalized.

Critical Force

The average distance from the threshold gives a simple relation for the critical force, namely $1-f_{c}={\overline {x}}$. Hence for the automata model we obtain:

$f_{c}=1-\int _{0}^{\infty }dxxP_{\text{stat}}(x)=1-{\frac {1}{2}}{\frac {\overline {\Delta ^{2}}}{\overline {\Delta }}}$

Exercise:

Let's assume an exponential distribution of the thresholds and show

• $P_{\text{stat}}(x)=e^{-x/{\overline {\Delta }}}/{\overline {\Delta }}$
• $f_{c}=1-{\overline {\Delta }}$

Avalanches or instability?

We consider an avalanche starting from a single unstable site $x_{0}=0$ and the sequence of sites more close to instabitity $x_{1}<x_{2}<x_{3}\ldots$. For each unstable block, all the blocks receive a random kick:

${\frac {\Delta _{1}}{(1+m^{2})L}},\quad {\frac {\Delta _{2}}{(1+m^{2})L}},\quad {\frac {\Delta _{3}}{(1+m^{2})L}},\ldots$

with $\Delta _{1},\Delta _{2},\Delta _{3},\ldots$ drwan from $g(\Delta )$ 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

$\int _{0}^{x_{1}}P_{w}(t)dt={\frac {1}{L}}$

Hence, for large systems we have

$x_{1}\sim {\frac {1}{LP_{w}(0)}},\;x_{2}\sim {\frac {2}{LP_{w}(0)}},\;x_{3}\sim {\frac {3}{LP_{w}(0)}},\ldots$

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

${\frac {\overline {\Delta }}{(1+m^{2})L}}\quad {\text{versus }}\quad {\frac {1}{P_{w}(0)L}}$

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 $1/(1+m^{2})$. Hence, the system is subcritical when $m>0$ and critical for $m=0$

Mapping to the Brownian motion

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

$\eta _{1}={\frac {\Delta _{1}}{(1+m^{2})L}}-x_{1},\;\eta _{2}={\frac {\Delta _{2}}{(1+m^{2})L}}-(x_{2}-x_{1}),\;\eta _{3}={\frac {\Delta _{3}}{(1+m^{2})L}}-(x_{3}-x_{2})\ldots$ $X_{n}=\sum _{i=1}^{n}\eta _{i}\quad \quad {\text{with}}\;{\overline {\eta _{i}}}={\frac {\overline {\Delta }}{L(1+m^{2})}}-{\frac {1}{LP_{w}(0)}}$

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 ${\frac {1}{LP_{w}(0)}}$ with ${\frac {1}{LP_{\text{stat}}(0)}}={\frac {\overline {\Delta }}{L}}$ we get $\;{\overline {\eta _{i}}}\sim -{\frac {m^{2}}{1+m^{2}}}{\frac {\overline {\Delta }}{L}}$. For small m, the random walk is only sliglty tilted. The avalanche distribution will be power law distributed with $\tau =3/2$ until a cut-off

$S_{\max }\sim m^{-4}$