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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