Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

Journal of Statistical Physics (Impact Factor: 1.28). 09/2009; 138(4-5). DOI: 10.1007/s10955-009-9901-3
Source: arXiv

ABSTRACT Adding sand grains at a single site in Abelian sandpile models produces beautiful but complex patterns. We study the effect of sink sites on such patterns. Sinks change the scaling of the diameter of the pattern with the number $N$ of sand grains added. For example, in two dimensions, in presence of a sink site, the diameter of the pattern grows as $\sqrt{(N/\log N)}$ for large $N$, whereas it grows as $\sqrt{N}$ if there are no sink sites. In presence of a line of sink sites, this rate reduces to $N^{1/3}$. We determine the growth rates for these sink geometries along with the case when there are two lines of sink sites forming a wedge, and its generalization to higher dimensions. We characterize one such asymptotic patterns on the two-dimensional F-lattice with a single source adjacent to a line of sink sites, in terms of position of different spatial features in the pattern. For this lattice, we also provide an exact characterization of the pattern with two sources, when the line joining them is along one of the axes. Comment: 27 pages, 17 figures. Figures with better resolution is available at

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Available from: Deepak Dhar, Jul 29, 2015
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    ABSTRACT: We consider patterns generated by adding large number of sand grains at a single site in an abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show proportionate growth. We study the robustness of these patterns against different types of noise, \textit{viz.}, randomness in the point of addition, disorder in the initial periodic configuration, and disorder in the connectivity of the underlying lattice. We find that the patterns show a varying degree of robustness to addition of a small amount of noise in each case. However, introducing stochasticity in the toppling rules seems to destroy the asymptotic patterns completely, even for a weak noise. We also discuss a variational formulation of the pattern selection problem in growing abelian sandpiles. Comment: 15 pages,16 figures
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    ABSTRACT: We study the patterns formed by adding N sand grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as N(1/d) for large N, in d dimensions. On the other hand, if sites with maximum stable height in the starting configuration form an infinite cluster, we get avalanches that do not stop. In this paper we describe our unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior: For any N, the avalanches are finite, but the diameter of the pattern increases as N(α), for large N, with 1/2<α≤1. Different values of α can be realized on different backgrounds, and the patterns still show proportionate growth. The noncompact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with α=1.
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