We discuss the properties of a self{organized critical forest{ re model which has been introduced recently. We derive scaling laws and de ne critical exponents. The values of these critical exponents are determined by computer simulations in 1 to 8 dimensions. The simulations suggest a critical dimension dc = 6 above which the critical exponents assume their mean{ eld values. Changing the lattice symmetry and allowing trees to be immune against re, we show that the critical exponents are universal.
We review the properties of the self-organized critical (SOC) forest-fire model. The paradigm of self-organized criticality refers to the tendency of certain large dissipative systems to drive themselves into a critical state independent of the initial conditions and without fine-tuning of the parameters.After an introduction, we define the rules of the model and discuss various large-scale structures which may appear in this system. The origin of the critical behavior is explained, critical exponents are introduced, and scaling relations between the exponents are derived. Results of computer simulations and analytical calculations are summarized. The existence of an upper critical dimension and the universality of the critical behavior under changes of lattice symmetry or the introduction of immunity are discussed. A survey of interesting modifications of the forest-fire model is given. Finally, several other important SOC models are briefly described.
We present the analytic solution of the self-organized critical (SOC) forest-fire model in one dimension proving SOC in systems without conservation laws by analytic means. Under the condition that the system is in the steady state and very close to the critical point, we calculate the probability that a string of n neighboring sites is occupied by a given configuration of trees. The critical exponent describing the size distribution of forest clusters is exactly τ = 2 and does not change under certain changes of the model rules. Computer simulations confirm the analytic results.
We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems. Rather, the system is composed of relatively homogeneous patches of different tree densities, leading to two qualitatively different types of fires: those that span an entire patch and those that don't. As the system size becomes smaller, the system contains less patches, and finally becomes homogeneous, with large density fluctuations in time.
We introduce a nonequilibrium percolation model which shows a selforganized critical (SOC) state and several periodic states. In the SOC state, the correlation length diverges slower than the system size, and the corresponding exponent depends non universally on the parameter of the model. The periodic states contain an infinite cluster covering only part of the system.
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