Fine Structure of Avalanches in the Abelian Sandpile Model

Source: arXiv


We study the two-dimensional Abelian Sandpile Model on a square lattice of
linear size L. We introduce the notion of avalanche's fine structure and
compare the behavior of avalanches and waves of toppling. We show that
according to the degree of complexity in the fine structure of avalanches,
which is a direct consequence of the intricate superposition of the boundaries
of successive waves, avalanches fall into two different categories. We propose
scaling ans\"{a}tz for these avalanche types and verify them numerically. We
find that while the first type of avalanches has a simple scaling behavior, the
second (complex) type is characterized by an avalanche-size dependent scaling
exponent. This provides a framework within which one can understand the failure
of a consistent scaling behavior in this model.

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Available from: Afshin Montakhab, Jan 30, 2013
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  • 01/1992; Addison-Wesley.
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    ABSTRACT: The gaps in the fossil record gave rise to the hypothesis that evolution proceeded in long periods of stasis, which alternated with occasional, rapid changes that yielded evolutionary progress. One mechanism that could cause these punctuated bursts is the recolonization of changing and deserted niches after mass extinction events. Furthermore, paleontological studies have shown that there is a power law relationship between the frequency of species extinction events and the size of the extinction impact. Power law relationships of this kind are typical for complex systems, which operate at a critical state between chaos and order, known as self-organized criticality (SOC). Based on this background, we used SOC to control the size of spatial extinction zones in a diffusion model. The SOC selection process was easy to implement and implied only negligible computational costs. Our results show that the SOC spatial extinction model clearly outperforms simple evolutionary algorithms (EAs) and the diffusion model (CGA). Further, our results support the biological hypothesis that mass extinctions might play an important role in evolution. However, the success of simple EAs indicates that evolution would already be a powerful optimization process without mass extinction, though probably slower and with less perfect adaptations
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