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 (mean field) model for the cellular automaton
Let's study the mean field version of the cellular automata introduced in the previous lecture. We introduce two approximations:
- Replace the Laplacian, which is short range, with a mean field fully connected interction
.
- The local threshold are all equal. In particular we set
.
As a consequence, in the limit , the statistical properties of the system are described by the distribution of the local stresses . For simplicity, instead of the stresses, we study the distance from threshold
Our goal is thus to determine their distribution , given their intial distribution, , and a value of .
Dynamics
Let's rewrite the dynamics with the new variables
- Drive: Increasing each point decreases its distance to threshold
.
As a consequence
- Instability 1: Stress drop The instability occurs when a point is at . Then, the point is stabilized (stress drop):
Increasing , a fraction of the blocks is unstable. Due to the stress drop, their distance to threshold becomes . Hence, one writes
- Instability 2: Stress redistribution The stress drop of a single block induces a stress redistribution where all blocks approach threshold.
The total stress drop is hence all points move to the origin of
part of them shifts, part of them become unstable... we can write
and finally:
Stationary solution
Increasing the drive the distribution converge to the fixed point:
- Determne using
- Show
which is well normalized.
Critical Force
The average distance from the threshold gives a simple relation for the critical force, namely . Hence for the automata model we obtain:
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} }
Avalanches or instability?
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) } . For each unstable block, all the blocks receive a kick. The mean value of the kick is
Is this kick able to destabilize another block? The equation setting the average position of the most unstable block is
Hence, for large systems we have
We expect three possibilities:
- if the mean kick, is smaller than the mean gap 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 \sim 1 /P_w(0)} , the system is subcritical and avalanches quickly stops.
- if the mean 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 \sim \overline{\Delta}/(1+m^2) } is equal to the mean gap , the system is critical and avalanches are power law distributed
- if the mean 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 \sim \overline{\Delta}/(1+m^2) } is larger of the mean gap 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 \sim 1 /P_w(0)} , the system is super-critical and avalanches are unstable.
Noe that in the stationary regime 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