Characterization of structural reorganization in rice piles.
Diagnostics applied to a rice-pile cellular automaton reveal different mechanisms producing power-law behaviors of statistical attributes of grains which are germane to self organised critical phenomena. The probability distributions for these quantities can be derived from two distinct random walk models that account for correlated clustered behavior through incorporating fluctuations in the number of steps in the walk. The first model describes the distribution for a spatial quantity, the resultant flight length of grains. This has a power-law tail caused by grains moving through a discrete, power-law distributed number of random steps of finite length. Developing this model into a random walk obtains distributions for the resultant flight length with characteristics similar to Lévy distributions. The second random walk model is devised to explain a temporal quantity, the distribution of "trapping" or "residence" times of grains at single locations in the pile. Diagnostics reveal that the trapping time can be constructed as a sum of "subtrapping times," which are described by a Lévy distribution where the number of terms in the sum is a discrete random variable accurately described by a negative binomial distribution. The infinitely divisible, two-parameter, limit distribution for the resultant of such a random walk is discussed, and describes a dual-scale power-law behavior if the number fluctuations are strongly clustered. The form for the distribution of transit times of grains results as a corollary.
Hopcraft
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Theoretical Mechanics Division, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom.
Jakeman
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Tanner
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