Transduction on Kadanoff Sand Pile Model Avalanches, Application to Wave Pattern Emergence

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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. The main interest of sand piles relies in their Self Organized Criticality (SOC), the property that a small perturbation | adding some sand grains | on a fixed configuration has uncontrolled consequences on the system, involving an arbitrary number of grain fall. Physicists L. Kadanoff et al inspire KSPM, a model presenting a sharp SOC behavior, extending the well known Sand Pile Model. 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 the \(D\!-\!1\) adjacent columns to the right if the difference of height between columns i and \(i\!+\!1\) is greater or equal to D. This paper develops a formal background for the study of KSPM fixed points. This background, resumed in a finite state word transducer, is used to provide a plain formula for fixed points of KSPM(3).