L-6: Difference between revisions
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<center><math> x_i = 1-\sigma_i | <center><math> x_i = 1-\sigma_i | ||
</math></center> | </math></center> | ||
The instability occurs when a block is at <math> x_i =0 </math> and is followed by its stabilization and a redistribution on all the blocks : | |||
<center> | |||
<math> | |||
\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} | |||
</math> | |||
</center> | |||
=== Dynamics === | === Dynamics === | ||
Let's | Our goal is thus to determine the distribution <math>P_w(x)</math> of all blocks, given their intial distribution, <math>P_0(x)</math>, and a value of <math> w </math>. | ||
Let's decompose in steps the dynamics | |||
* <Strong> Drive:</Strong> Increasing <math>w \to w + dw</math> each | * <Strong> Drive:</Strong> Increasing <math>w \to w + dw</math> each block decreases its distance to threshold | ||
<center><math> x_i \to x_i - m^2 dw </math></center>. | <center><math> x_i \to x_i - m^2 dw </math></center>. | ||
As a consequence | As a consequence | ||
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* <Strong> | * <Strong> Stabilization : </Strong> A fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math>m^2 d w P_w(0) \to m^2 d w P_w(0) g(x) </math>. Hence, one writes | ||
<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center> | <center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center> | ||
* <Strong> Redistribution </Strong> | * <Strong> Redistribution </Strong> The total drop on the force acting on the unstable blocks <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> per unit length. This drop is partially compensated by the redistribution. The force acting on all points is increased: | ||
<center> <math> x_i \to x_i - m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center> | |||
Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write | |||
The total drop | |||
This drop | |||
<center> <math> m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center> | |||
Again, most of the distribution will be driven to instability while a | |||
<center> <math>\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] </math> </center> | <center> <math>\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] </math> </center> | ||
and finally: | and finally: |
Revision as of 20:49, 8 March 2025
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
The instability occurs when a block is at and is followed by its stabilization and a redistribution on all the blocks :
Dynamics
Our goal is thus to determine the distribution of all blocks, given their intial distribution, , and a value of . Let's decompose in steps the dynamics
- Drive: Increasing each block decreases its distance to threshold
.
As a consequence
- Stabilization : A fraction of the blocks is unstable. The stabilization induces the change . Hence, one writes
- Redistribution The total drop on the force acting on the unstable blocks per unit length. This drop is partially compensated by the redistribution. The force acting on all points is increased:
Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write
and finally:
Stationary solution
Increasing the drive the distribution converge to the fixed point:
- Determine 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
Avalanches or instability?
Given the initial condition and , the state of the system is described by . 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 , 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 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
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 steps is independent on the jump disribution and for a large number of steps becomes . Hence, the distribution avalanche size is
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 we get . For small m, the random walk is only sliglty tilted. The avalanche distribution will be power law distributed with until a cut-off
Bienaymé Galton Watson process
A time appears as infected individual which dies with a rate and branches with a rate . On average, each infection generates in average new ones. Real epidemics corresponds to .
At time , the infected population is , while the total infected population is
Our goal is to compute and we introduce its Laplace Transform:
. Note that the normalization imposes .
- Evolution equation: Consider the evolution up to the time as a first evolution from to and a following evolution from to . Derive the following equation for
which gives
- Critical case: the stationary solution: Let's set and to recover the results of the mean field cellular automata. In the limit the total population coincides with the avalanche size, . The Laplace transform of is
which gives
with
- Critical case: Asymptotics: We want to predict the power law tail of the avalanche distribution . Taking the derivative with respect to we have
and conclude that and
Hence we find back our previous result