Communications in Nonlinear Science and Numerical Simulation
Reprint of Infinity computations in cellular automaton forest-fire model☆
Introduction
Recently a new applied point of view on infinite and infinitesimal numbers has been introduced in [1], [19], [24], [26]. The new approach does not use Cantor’s ideas (see [6]) and describes infinite and infinitesimal numbers that are in accordance with the principle ‘The whole is greater than the part’ (Euclid’s Common Notion 5). It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of computers – the Infinity Computer – introduced in [20], [21], [27], [28]. In our previous paper [2] we applied the new computational tools to study percolation phase transition. It has been established that in an infinite system phase transition point is not really a point as with respect of traditional approach. We showed that in light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on the “microscope” we use this interval could be regarded as either finite or infinite or infinitesimal interval. Using new approach we observed that in vicinity of percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional consideration. Moreover, we have now a tool to distinguish those infinite clusters. In particular, we can distinguish spanning infinite clusters from embedded infinite clusters.
In this paper, we are going to apply the new arithmetic to a cellular automaton forest-fire model [10], [11] which is tightly connected with the percolation methodology and in some sense combines the dynamic and the static percolation problems. Forest-fire model elegantly represents the simplest example of a model that under certain conditions exhibits critical fluctuations. Actually, community that deals with the forest-fire model and its applications, distinguishes between two versions of the model: real forest-fire model where fire catches adjacent trees in the forest in the step by step manner and a simplified version with the instantaneous combustion [10], [12]. Using the new approach we show that in both situations we deal with the same model but with different time resolution. We observe that depending on “microscope” we use the same cellular automaton forest-fire model reveals either instantaneous forest combustion or step by step firing.
Another interesting feature of the forest-fire model is that the model is critical when driven in a certain limit. That is, critical behavior will occur in the limit of slowly growing trees. Though slow-growing, the trees must grow quickly compared to the time interval between spontaneously ignited fires, i.e., the ratio between the ignition probability and the growth probability should be moved toward zero. It should be emphasized that the limits discussed imply infinite model system size, so-called thermodynamic limit. Finite model size limit results in some difficulties during cellular automaton forest-fire model implementation in numerical experiment. The new computational approach proposed recently in [19], [24] allows us to overcome this difficulties. As far as we choose infinitesimal trees growing rate and infinitesimal ratio between the ignition probability and the growth probability we determine the measure or extent of the system size infinity that provides the criticality of the system dynamics. By means of the new approach we derive correspondent inequalities for grosspowers that control trees growing rate infinitesimality, infinitesimal ignition probability and the system size infinity.
The outline of the paper is as follows. In Section 2 we introduce the new approach methodology that allows one to write down different finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework and to execute numerical computations with all of them. Then in Section 3 we briefly remind basic features of the percolation phase transition and summarize some features of infinity percolation cluster in terms of infinity computations. In Section 4 we apply the new arithmetic to the cellular automaton forest-fire model. In the final section, the application results are summarized and discussed.
Section snippets
Methodology
In this section, we give a brief introduction to the new methodology that can be found in a rather comprehensive form in [24], [26], [30] downloadable from [21] (see also the monograph [19] written in a popular manner). A number of applications of the new approach can be found in [7], [8], [16], [17], [22], [23], [25], [27], [28], [29], [30], [31], [32], [34], [35]. We start by introducing three postulates that will fix our methodological positions (having a strong applied character) with
Geometric phase transition
Consider a d-dimensional hypercubic lattice, where each site is either occupied randomly with a probability p or empty with probability . Occupied and empty sites may stand for very different physical properties [4], [5], [13], [33]. For purposes of further inquiry, let us assume that the occupied sites are trees, and that combustion front can flow only between nearest neighbor occupied sites. At a low concentration p, the occupied sites are either isolated or form small clusters of nearest
Forest fires model
In this section, we are going to apply the new arithmetic to a self-organized critical forest-fire model (see [10], [11]) which is tightly connected with the percolation methodology and in some sense combines the dynamic and the static percolation problems. Let us assume, that we examine the forest fire model on a d-dimensional hypercubic lattice with the lattice spacing a and on the linear scale . Then, the lattice contains the infinite number of sites . A lattice site can be in one
A brief conclusion
In this paper, it has been shown that infinite and infinitesimal numbers introduced in [19], [24], [26] allow us to obtain exact numerical results instead of traditional asymptotic forms at different points at infinity. We consider a number of traditional models related to the percolation theory using the new computational methodology. It has been shown that the new computational tools allow one to create new, more precise models of percolation and to study the existing models more in detail.
Acknowledgments
This work was supported in part by Grants from the Government of the Russian Federation (contracts Nos. 14.B25.31.0023; 11.G34.31.0048) and by the Russian Foundation for Basic Research (project No. 13-01-97063).
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The publisher would like to inform the readership that this article is a reprint of a previously published article. An error occurred which resulted in the publication of this article in a wrong issue. As a consequence, the publisher would like to make this reprint available for the reader’s convenience and for the continuity of the papers involved in the Special Issue. For citation purposes, please use the original publication details; Communications in Nonlinear Science and Numerical Simulation. Volume 20/3, March 2015, Pages 861-870. The publisher sincerely apologizes to the readership.