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

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).