Randmoness and Step-like Distribution of Pile Heights in Avalanche Models

TL;DR

This paper introduces a parametric family of sand-pile models that interpolate between deterministic and stochastic relaxation, revealing a step-like distribution of pile heights and analyzing their properties through a spectral approach.

Contribution

It develops a unified model capturing the crossover between deterministic and stochastic relaxation in sand-piles, with detailed spectral analysis of height distributions.

Findings

Height densities tend to one another as the parameter increases.

Distributions exhibit step-like behavior contrasting with peaked shapes.

Spectral approach effectively distinguishes between relaxation types.

Abstract

The paper develops one-parametric family of the sand-piles dealing with the grains' local losses on the fixed amount. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean height of the pile is destined to describe the crossover. The height's densities corresponding to the models with relaxation of the both types tend one to another as the parameter increases. These densities follow a step-like behaviour in contrast to the peaked shape found in the models with the local loss of the grains down to the fixed level [S. Lubeck, Phys. Rev. E, 62, 6149, (2000)]. A spectral approach based on the long-run properties of the pile height considers the models with deterministic and random relaxation more accurately and distinguishes the both cases up to admissible parameter values.