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.

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_1,\Delta_2,\Delta_3, \ldots } drwan from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\Delta) } Are these kick able to destabilize other blocks?

Given the initial condition and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w } , the state of the system is described by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_w(x) } . From the extreme values theory we know the equation setting the average position of the most unstable block is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^{x_1} P_w(t) dt =\frac{1}{L} }

Hence, for large systems we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)}, \ldots }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\overline{\Delta}}{(1+m^2)L} \quad \text{versus }\quad \frac{1}{ P_w(0) L} }

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/(1+m^2) } . Hence, the system is subcritical when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m>0 } and critical for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=0 }

Mapping to the Brownian motion

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_0=0} . Under these hypothesis the Sparre-Andersen theorem state that the probability that the random walk remains positive for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} steps is independent on the jump disribution and for a large number of steps becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(n) \sim \frac{1}{\sqrt{\pi n}}} . Hence, the distribution avalanche size is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau=3/2}

• Stationary regime: Replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{LP_w(0)}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{LP_{\text{stat}}(0)} = \frac{\overline{\Delta}}{L} } we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \; \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau=3/2} until a cut-off

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\max} \sim m^{-4}}