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

TL;DR

This paper develops a formal framework for analyzing fixed points in the Kadanoff Sand Pile Model, particularly providing a formula for KSPM(3), enhancing understanding of self-organized criticality in these systems.

Contribution

It introduces a formal background and a finite state transducer approach to characterize fixed points in KSPM, especially for D=3, advancing theoretical understanding.

Findings

Provided a formal background for KSPM fixed points

Developed a finite state transducer model

Derived a formula for fixed points in KSPM(3)

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. The main interest of sand piles relies in their {\em 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 {\em et al} inspire KSPM, a model presenting a sharp SOC behavior, extending the well known {\em 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).