Pattern formation in growing sandpiles

EPL (Europhysics Letters) (Impact Factor: 2.27). 08/2008; DOI: 10.1209/0295-5075/85/48002
Source: arXiv

ABSTRACT Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern. Comment: 5 pages, 4 figures, submitted to Phys. Rev. Lett


Available from: Deepak Dhar, Feb 04, 2013
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    ABSTRACT: This book opens with an introduction to Abelian sandpile models, such that the reader can familiarize himself with the construct and some of the notation that is used throughout the book, as well as with the problems that will be tackled in the subsequent chapters. The third chapter provides new results with regard to the algebraic structures of ABMs, while the fourth chapter covers analytical results on the identity configuration of the ABM and the last one deals with standard experimental protocols that one can devise for the operators introduced in the third chapter. I my opinion, books are conceived to give their readers a comprehensive overview of the state of the art in a given field, whereas the main goal of a scientific paper is to disseminate new research results on, mostly, very specific topics. Typically, only those research results that stand out throughout the course of time are considered important enough to end up in books, as such becoming part of the scientific knowledge. In that respect, I do not think that there is much added value in reading this book, especially if one is looking for a comprehensive overview of the state of the art in the field of ABMs, as a consequence of the fact that it merely constitutes the bounded counterpart of the author’s papers that were already published in dedicated journals. Hence, if one wants to know more about the advances made possible in the field by this author, one should consult the corresponding papers. Aside from an introduction to ABMs in the second chapter it does not contain much additional information. I checked the text of the papers myself and it appears that the text in the book is largely literally the same as the one in the papers. Besides, a book should be proofread for grammar and spelling, something that hasn’t been done in the case of this work.
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    ABSTRACT: We study the growing patterns in the rotor-router model formed by adding $N$ walkers at the center of a $L \times L$ two-dimensional square lattice, starting with a periodic background of arrows, and relaxing to a stable configuration. The pattern is made of large number of triangular regions, where in each region all arrows point in the same direction. The square circumscribing the region, where all the arrows have been rotated atleast one full circle, may be considered as made up of smaller squares of different sizes, all of which grow linearly with $N$, for $ 1 \ll N < 2 L$. We use the Brooks-Smith-Stone-Tutte theorem relating tilings of squares by smaller squares to resistor networks, to determine the exact relative sizes of the different elements of the asymptotic pattern. We also determine the scaling limit of the function describing the variation of number of visits to a site with its position in the pattern. We also present evidence that deviations of the sizes of different small squares from the linear growth for finite $N$ are bounded and quasiperiodic functions of $N$.
    Journal of Statistical Mechanics Theory and Experiment 12/2013; 2014(11). DOI:10.1088/1742-5468/2014/11/P11030 · 2.06 Impact Factor
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    ABSTRACT: An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. In this paper, we review our recent work on patterns formed by adding $N$ grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. We give several examples of such patterns. We discuss the special cases where the asymptotic pattern can be determined exactly. The effect of noise in the background or in the rules on the patterns is also discussed.
    Journal of Statistical Mechanics Theory and Experiment 10/2013; DOI:10.1088/1742-5468/2013/11/P11006 · 2.06 Impact Factor