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 {\em et al} inspire KSPM, extending the well known {\em 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 $i+1,i+2,\dots,i+D\!-\!1$ 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.