Reprint of Infinity computations in cellular automaton forest-fire model

Abstract

Recently a number of traditional models related to the percolation theory has been considered by means of a new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The whole is greater than the part’ (Euclid’s Common Notion 5). Here we apply the new arithmetic to a cellular automaton forest-fire model which is connected with the percolation methodology and in some sense combines the dynamic and the static percolation problems and under certain conditions exhibits critical fluctuations. It is well known that there exist two versions of the model: real forest-fire model where fire catches adjacent trees in the forest in the step by step manner and simplified version with instantaneous combustion. Using new approach we observe that in both situations we deal with the same model but with different time resolution. We show that depending on the “microscope” we use the same cellular automaton forest-fire model reveals either instantaneous forest combustion or step by step firing. By means of the new approach it was also observed that as far as we choose an infinitesimal tree 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. Correspondent inequalities for grosspowers are derived.