L-6

Avalanches and Bienaymé-Galton-Watson process

Goal: We solve the mean field version of the cellular automaton, derive its avalanche statistics and make a connection with the Bienaymé-Galton-Watson process used to describe an epidemic outbreak.

Fully connected model foor the cellular automata (mean field)

Let's study the mean field version of the cellular automata introduced in the previous lecture.

• The elastic coupling is with all neighbours

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

.

• The local random threshold are all equal:

$\sigma _{i}^{th}=1,\quad \forall i$

.

Instead of following the evoluion of the $\sigma _{i}$, it is useful to introduce the distance from threshold

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

Hence, an unstable point, $x_{i}<0$, is stabilized to a value $x>0$ drawn from $g(x)$. The stress redistribution induced on each bloch is ${\frac {1}{L}}{\frac {x}{1+m^{2}}}$

In the limit $L\to \infty$ we define the distribution $P_{w}(x)$ and write its evolution equation.

• Drive: Changing $w\to w+dw$ gives $P_{w+dw}(x)=P_{w}(x+dw)\sim P_{w}(x)+m^{2}dw\partial _{x}P_{w}(x)$
• Instability: This shift is stable far from the origin, however for a fraction $m^{2}dwP_{w}(0)$ of the points of the interface is unstable. Due to the stress drop, their distance to instability will be $m^{2}dwP_{w}(0)g(x)$. Hence, one writes

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

• Stress redistribution: as a consequence all points move to the origin of

$m^{2}dwP_{w}(0){\frac {\overline {x}}{1+m^{2}}}$

where ${\overline {x}}=\int dxxg(x)$

• Avalanche: Let us call we can write

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

and finally:

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

Stationary solution

Increasing the drive the distribution converge to the fixed point:

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

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

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

• Show

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