Interpretation of percolation in terms of infinity computations

In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives...

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Bibliographic Details
Published in:Applied mathematics and computation 2012-04, Vol.218 (16), p.8099-8111
Main Authors: Iudin, D.I., Sergeyev, Ya.D., Hayakawa, M.
Format: Article
Language:English
Subjects:
Clusters
Computation
Computer simulation
Infinite sets
Infinity
Intervals
Mathematical analysis
Mathematical models
Percolation
Percolation cluster
Phase transition
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Summary:In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer – the Infinity Computer – introduced recently in [18]. The new approach does not contradict Cantor. In contrast, it can be viewed as an evolution of his deep ideas regarding the existence of different infinite numbers in a more applied way. Site percolation and gradient percolation have been studied by applying the new computational tools. It has been established that in an infinite system the phase transition point is not really a point as with respect of traditional approach. In light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on “microscope” we use this interval could be regarded as finite, infinite and infinitesimal short 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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2011.11.044