Avalanche Structure in the Kadanoff Sand Pile Model

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AbstractSand pile models are dynamical systems emphasizing the phenomenon of Self Organized Criticality (SOC). From N stacked grains, iterating evolution rules leads to some critical configuration where a small disturbance has deep consequences on the system, involving numerous steps 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 with parameter 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. We propose an iterative study of KSPM evolution where one single grain addition is repeated on a heap of sand. The sequence of grain falls following a single grain addition is called an avalanche. From a certain column precisely studied for D = 3, we provide a plain process describing avalanches. We hope that this process is a first stone toward the study of KSPM fixed points structure.Keywordsdiscrete dynamical systemself-organized criticalitysand pile model