L-6

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

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