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
The stabilization of the unstable blocks induce a drop of the force per unit length
\
- Redistribution 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
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?
We consider an avalanche starting from a single unstable site
and the sequence of sites more close to instabitity
. For each unstable block, all the blocks receive a random kick:
with
drwan from
Are these kick able to destabilize other blocks?
Given the initial condition and
, the state of the system is described by
. From the extreme values theory we know the equation setting the average position of the most unstable block is
Hence, for large systems we have
Hence we need to compare the mean value of the kick with the mean gap between nearest unstable sites:
Note that
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
. Hence, the system is subcritical when
and critical for
Mapping to the Brownian motion
Let's define the random jumps and the associated random walk
An avalanche is active until
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
. 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
- Stationary regime: Replacing
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