The Bak-Chen-Tang Forest Fire Model Revisited

Since Self-Organised Criticality (SOC) was introduced in the 1987 both the nature of the self-organisation and of the criticality remains controversial. Recent observations on rain precipitation and on brain activity suggest that real systems may dynamically wander about in the vicinity of criticality rather than tune to a critical point. We use computer simulations to study the Drossel-Schwable forest-fire model of SOC and find that it exhibits behaviour similar to that foun...

We present high statistics Monte Carlo results for the Drossel-Schwabl forest fire model in 2 dimensions. They extend to much larger lattices (up to $65536\times 65536$) than previous simulations and reach much closer to the critical point (up to $\theta \equiv p/f = 256000$). They are incompatible with all previous conjectures for the (extrapolated) critical behaviour, although they in general agree well with previous simulations wherever they can be directly compared. Inste...

The Bak-Chen-Tang forest fire model was proposed as a toy model of turbulent systems, where energy (in the form of trees) is injected uniformly and globally, but is dissipated (burns) locally. We review our previous results on the model and present our new results on the statistics of the higher-order moments for the spatial distribution of fires. We show numerically that the spatial distribution of dissipation can be described by Log-Poisson statistics which leads to extende...

We introduce a novel approach to study the critical behavior of equilibrium and non-equilibrium systems which is based on the concept of an instantaneous correlation length. We analyze in detail two classical statistical mechanical systems: the XY model and the Ising model, and one of the prototype models of Self-Organized Criticality: the forest fire model (FFM). The proposed method can both capture the critical behavior of the XY model and the Ising model and discriminate b...

We re-examine a two-dimensional forest-fire model via Monte-Carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two-point correlation function of trees. We check resp. improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode. We also investigate the possibility of describing the critical poi...

We present a brief review of power laws and correlation functions as measures of criticality and the relation between them. By comparing phenomenology from rain, brain and the forest fire model we discuss the relevant features of self-organisation to the vicinity about a critical state. We conclude that organisation to a region of extended correlations and approximate power laws may be behaviour of interest shared between the three considered systems.

The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, as forest-fires or earthquake automata, is scrutinized in this paper. By using three different levels of description - (i) a simple mean field, (ii) a more detailed mean-field description in terms of a (self-organized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation-, it is shown on general grounds that non-co...

Recently, large-scale cascading failures in complex systems have garnered substantial attention. Such extreme events have been treated as an integral part of the self-organized criticality (SOC). Recent empirical work has suggested that some extreme events systematically deviate from the SOC paradigm, requiring a different theoretical framework. We shed additional theoretical light on this possibility by studying financial crisis. We build our model of financial crisis on the...

We investigate a forest-fire model with the density of empty sites as control parameter. The model exhibits three phases, separated by one first-order phase transition and one 'mixed' phase transition which shows critical behavior on only one side and hysteresis. The critical behavior is found to be that of the self-organized critical forest-fire model [B. Drossel and F. Schwabl, Phys. Rev. Lett. 69, 1629 (1992)], whereas in the adjacent phase one finds the spiral waves of th...

The one-dimensional forest-fire model including lightnings is studied numerically and analytically. For the tree correlation function, a new correlation length with critical exponent \nu ~ 5/6 is found by simulations. A Hamiltonian formulation is introduced which enables one to study the stationary state close to the critical point using quantum-mechanical perturbation theory. With this formulation also the structure of the low-lying relaxation spectrum and the critical behav...