LBan-V

Fully connected model with parabolic potential

An alternative is to control the displacement of the interface. To achieve this, we introduce a parabolic potential which attracts each block to position $w$ with a spring constant $m^{2}$. The local force

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

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. For simplicity, instead of the stresses, we study the distance from threshold

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

The instability occurs when a block is at $x_{i}=0$ and is followed by its stabilization and a redistribution on all the blocks :

${\begin{cases}x_{i}=0\to x_{i}=\Delta &(stabilization)\\x_{j}\to x_{j}-{\frac {1}{L}}{\frac {\Delta }{1+m^{2}}}&(redistribution)\\\end{cases}}$

Dynamics

Our goal is thus to determine the distribution 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)} of all blocks, given their intial distribution, 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_0(x)} , and a value of $w$. Let's decompose in steps the dynamics

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

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_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 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^2 d w P_w(0) } of the blocks is unstable. The stabilization induces the change 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^2 d w P_w(0) \to m^2 d w P_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

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^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} }

\

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

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_i \to x_i - m^2 dw P_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:

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 \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:

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 0 = \partial_x P_{\text{stat}}(x) + P_{\text{stat}}(0) g(x) }

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

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= \int_0^\infty dx \, P_{\text{stat}}(x)= - \int_0^\infty dx \, x \partial_x P_{\text{stat}}(x) }

• Show

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_{\text{stat}}(x)= \frac{1}{\overline{\Delta}} \int_x^\infty g(z) d z }

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:

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 f_c= 1- \int_0^\infty d x x P_{\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

• 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_{\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 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 } and the sequence of sites more close to instabitity 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< 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 $\Delta _{1},\Delta _{2},\Delta _{3},\ldots$ 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 $P_{w}(x)$. 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:

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

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 $n$ 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

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}}