Kadanoff sand pile model. Avalanche structure and wave shape

Abstract

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.

Strong Emergence of Wave Patterns on Kadanoff Sandpiles

“…A series of previous works ( [30,31,33]) lead us to a similar result, but only for the smallest "new" parameter p = 2 (the case p = 1 is the well known sandpile model), using exclusively arguments of combinatorics. However, for the general case, we introduce a completely different approach.…”

“…An interesting property of an avalanche is the absence of hole from an index the electronic journal of combinatorics 24(2) (2017), #P2.4 l, which tells that there exists an m such that for all i with l i m, we have i ∈ s k for all i with m < i, we have i / ∈ s k namely, from column l, a set of consecutive columns is fired, and nothing else. We say that an avalanche s k is dense starting from an index l when s k contains no hole i with i l. We studied the structure of avalanches in [30,31,33], and saw that this property leads to important regularities in successive fixed points. It induces a kind of "pseudo linearity" on avalanches, it somehow "breaks" the criticality of avalanche's behavior and let them flow smoothly along the sandpile.…”

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

“…Dans une série d'articles [19,20,21,23,25] nous avons conclu une caractérisation asymptotique des points fixes π(N ) pour les modèles de piles de sable Kadanoff, dont les règles sont de la forme (1, . .…”

“…. , 1)) dans [19,20,21,23,25], et des premierś eléments de leur généralisation ontété présentés dans [22,24]. Ces développements utilisent la convergence de la représentation ∆ 2 v des points fixes, observable sur les simulations de la Section 3.…”

“…En conséquence, les configurations stables obtenues présentent un segment irrégulier (où les grains sont ajoutés), suivi d'un segment très ordonné ayantémergé de la dynamique. Ces comportements ontété démontrés pour les modèles de piles de sable Kadanoff, par l'introduction d'une technique de preuve mêlant deséléments d'algèbre linéaire pour l'apparition de régularités (notion de convergence), et combinatoire pour la description fine des motifs réguliers [19,20,21,23,25].…”