Self-organized criticality and the forest-fire model
Self-organized criticality and the forest-fire model
![](http://perso.u-cergy.fr/~ahonecker/bs/software/ffm2d/forest2d.gif)
Self-organized criticality and the forest-fire model
The concept of
`self-organized crititcality' has been introduced by P. Bak and
collaborators to explain the occurence of power laws in nature
(1/f noise, magnitude of earthquakes, ...). Recent
experiments with quasi-one-dimensional piles of rice
(Frette et.al., nature 379 (1996) 49-52) do indeed
show power laws whose existence, however, depends on such
microscopic details as the aspect ratio of the grains of rice.
One of the toy models proposed for self-organized criticality is a
forest-fire model. We have investigated this model in one
and two spatial dimensions.
One of our main observations is that this model has two different
length scales whose critical exponents are distinct: One related to
clusters and one to the usual correlation of occupancy of sites
anywhere in the system. The latter quantity is frequently not
examined in the context of models of self-organized criticality
(sometimes because these two distinct quantities are confused
with each other). This neglect of the (in the context of equilibrium
statistical mechanics) well-known usual correlation functions
is surprising and one should pay more attention to them. They should
also be simple to measure in experiments and could help to
decide which model describes the experiments best.
A second result is that power laws in cluster sizes can easily
be obtained after discarding the spatial structure and then looking
only at a distribution of global densities of trees. In one
dimension, this describes all correlation functions at the
critical point correctly, while in two dimensions one obtains just
qualitative (but no quantitive) agreement for the cluster-size
distribution. The off-critical
behaviour of the full model is not correctly described by
such global models, nor is the relaxational behaviour.
Let us now take a closer look at the one-dimensional model:
All exponents known previously are integers.
We have performed Monte-Carlo simulations of the
two-point correlation function which had not been studied before.
This revealed a new length scale with exponent \nu_T \approx 5/6. We
have then introduced a Hamiltonian formulation of the model and used
quantum-mechanical perturbation theory in order to study the stationary
state and find all correlation functions at the critical point. This
stationary state is invariant under permutation of the lattice sites
which can be exploited to introduce a simplified model involving
only the total number of trees (see above).
We have also studied the relaxation spectrum of the full model numerically
and obtained two further new critical exponents.
We have also simulated the two-dimensional forest-fire model.
Here one also finds two different length scales with
different exponents related to cluster quantities and the two-point
correlation function. We also checked or improved a few other exponents
and looked at the relaxational behaviour which had not been
investigated in this manner before. Finally, we have also extracted
a global density distribution from simulations.
Such global density distributions can then be used as the basis
of yet another simulation in order to compute the exponent
of the cluster-size distribution (compare also the remarks above).
Another summary of some of our results can be found on
this poster.
If you would like to know more about the one-dimensional case,
take a look at
this paper.
More details about our results in two dimensions can be found in
this paper.
If you would like to do some simulations and see nice pictures of the
two-dimensional model, look
here
for a program including X11 support.
September, 12th, 1996
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