Avalanche Structure in the Kadanoff Sand Pile Model

TL;DR

This paper analyzes the avalanche behavior in the Kadanoff Sand Pile Model, providing a detailed process description for avalanches, especially for the case D=3, to understand self-organized criticality.

Contribution

It offers a new iterative analysis of KSPM evolution and a detailed avalanche process description for D=3, extending understanding of SOC in sand pile models.

Findings

Describes avalanche process for D=3 in KSPM

Provides a plain process for avalanche analysis

Enhances understanding of SOC in sand pile models

Abstract

Sand 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.