Forest fires and other examples of self-organized criticality

Abstract

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.

Broken scaling in the forest-fire model

“…The value of τ * is close to (but not within the error of) the exponent found in the literature, τ = 2.14(3) [3,4] (τ = 2.15(2) in [16], τ = 2.159(6) in [6]), which is shown in the same figure for comparison. However, it is impossible to force the minima (see the down pointing marks in Fig.…”

“…The model has been described several times and in great detail elsewhere [1,3,4]. Therefore, the description presented here is rather succinct.…”

“…3) to the same height while maintaining the constraint that the maxima remain aligned, i.e. these , the other exponents are from literature, namely 2.14(3) in [3,4] and 223/91 ≈ 2.45 in [12]. Since it was impossible to relate these exponents to any property of the data, the exact position of the lines associated with them was chosen arbitrarily.…”

Critical behaviour of irreversible reaction systems

“…Some typical examples include the spreading of epidemics through a population, the propagation of rumors in a society [30], the spreading of fire through a forest [31,32], coexistence with extinction transitions in prey-predator systems [33] and, of course, catalytic and autocatalytic chemical reactions. From a more general point of view, absorbing states are expected to occur in situations where some quantity of interest can proliferate or die out ( e.g the fire in the forest), without any possibility of spontaneous generation (e.g due to the rays of an electrical storm).…”

“…Among others, KPZ and EW universality classes are the most frequently found in both experiments and models [67], including electrochemical deposition, polycrystalline thin-film growth, fire-front propagation, etc. [30,31,32,33].…”