Parallel transient time of one-dimensional sand pile

Durand-Lose, Jérôme

doi:10.1016/s0304-3975(97)00073-x

Cited by 18 publications

(21 citation statements)

References 4 publications

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“…The evolution starts from the initial configuration h where h 0 " N and h i " 0 for i ą 0, and in the classical sandpile model a grain falls from column i to column i`1 if and only if the height difference h i´hi`1 ą 1. One-dimensional sandpile models were well studied in recent years [11,4,12,5,25,22,6]. Kadanoff et al proposed a generalization of classical models in which a fixed parameter p denotes the number of grains falling at each step [17].…”

Section: Kadanoff Sandpile Model (Kspm)mentioning

confidence: 99%

Emergence of Wave Patterns on Kadanoff Sandpiles

Perrot

Rémila

2014

LATIN 2014: Theoretical Informatics

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Emergence is a concept that is easy to exhibit, but very hard to formally handle. This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sandpile Model is a discrete dynamical system describing the evolution of a finite number of stacked grains -as they would fall from an hourglass-to a stable configuration (fixed point). Grains move according to the repeated application of a simple local rule until reaching a fixed point. The main interest of the model relies in the difficulty of understanding its behavior, despite the simplicity of the rule. In this paper we prove the emergence of wave patterns periodically repeated on fixed points. Remarkably, those regular patterns do not cover the entire fixed point, but eventually emerge from a seemingly highly disordered segment. The proof technique we set up associated arguments of linear algebra and combinatorics, which interestingly allow to formally state the emergence of regular patterns without requiring a precise understanding of the chaotic initial segment's dynamic.

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Section: Kadanoff Sandpile Model (Kspm)mentioning

confidence: 99%

Emergence of Wave Patterns on Kadanoff Sandpiles

Perrot

Rémila

2014

LATIN 2014: Theoretical Informatics

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“…We first prove it restricting ourselves on words of L Given a finite word v on L, we define the maximal height g(v) = max{|h(v ′ )| |v ′ prefix of v}. The previous lemma gives the result g(t(v)) ≤ 1 + g(v) 4 . We can now use a trick to get the expected result.…”

mentioning

confidence: 99%

Kadanoff sand pile model. Avalanche structure and wave shape

Perrot

Rémila

2013

Theoretical Computer Science

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Sand pile models are dynamical systems describing the evolution from N stacked grains to a stable configuration. It uses local rules to depict grain moves and iterate it until reaching a fixed configuration from which no rule can be applied. Physicists L. Kadanoff et al inspire KSPM, extending the well known Sand Pile Model (SPM). In KSPM(D), we start from a pile of N stacked grains and apply the rule: D−1 grains can fall from column i onto columns i + 1, i + 2, . . . , i + D−1 if the difference of height between columns i and i+1 is greater or equal to D. Toward the study of fixed points (stable configurations on which no grain can move) obtained from N stacked grains, we propose an iterative study of KSPM evolution consisting in the repeated addition of one grain on a heap of sand, triggering an avalanche at each iteration. We develop a formal background for the study of avalanches, resumed in a finite state word transducer, and explain how this transducer may be used to predict the form of fixed points. Further precise developments provide a plain formula for fixed points of KSPM(3), showing the emergence of a wavy shape.

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“…Ces grains se déplacent entre les cases du tableau selon des règles d'évolution comme celles décrites par les Figures 1 et 2. Une attention particulière aété portée aux configurations obtenues en partant d'une unique colonne contenant un nombre fini de grains [8,9,11,6,17,26,27,13,14,10,7,16], amenant des développements mathématiques intéressants pour eux-même, et motivés par troiséléments :…”

Section: Introductionunclassified

Piles de sable décroissantes 1D. Classification expérimentale d'émergences

Perrot¹,

Rémila²

2015

Techniques et sciences informatiques

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RésuméFrançais. La dynamique des modèles de piles de sable décroissantes en une dimension laisse apparaître des motifs réguliers après une phase d'évolutionà l'allure désordonnée. Dans cet article, nous présentons les résultats de simulations mettant en avant des phénomènes d'émergence entretenus par différents mécanismes. La définition des modèles est argumentée, ses premières propriétés sont démontrées, et les observations sont discutées autour de la formulation de deux conjectures. Mots clés. Piles de sable, points fixes,émergence. English.The dynamics of decreasing sandpile models in one dimension lets regular patterns appear after a seemingly unordered phase of evolution. In this article, we present simulation results exhibiting emergence phenomena sustained by different mechanisms. The definition of decreasing sandpile models is argued, its main properties are demonstrated, and the observations are discussed around the statement of two conjectures.

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Parallel transient time of one-dimensional sand pile

Cited by 18 publications

References 4 publications

Emergence of Wave Patterns on Kadanoff Sandpiles

Emergence of Wave Patterns on Kadanoff Sandpiles

Kadanoff sand pile model. Avalanche structure and wave shape

Piles de sable décroissantes 1D. Classification expérimentale d'émergences

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