Residence Time Distribution of Sand Grains in the 1-Dimensional Abelian Sandpile Model

TL;DR

This paper investigates the residence time distribution of sand grains in a 1D abelian sandpile model, revealing exponential decay and a specific scaling form for large lattice sizes.

Contribution

It provides numerical analysis of residence time distribution and uncovers a novel scaling behavior with distinct exponents in the 1D abelian sandpile model.

Findings

Distribution decays exponentially with T/L^2

Distribution exhibits a scaling form with different exponents a and b

Numerical calculation of coefficient K_L up to L=150

Abstract

We study the probability distribution of residence time, $T$, of the sand grains in the one dimensional abelian sandpile model on a lattice of $L$ sites, for $T<<L^2$ and $T>>L^2$. The distribution function decays as $\exp(-\frac{K_LT}{L^2})$. We numerically calculate the coefficient $K_L$ for the value of $L$ upto 150 . Interestingly the distribution function has a scaling form $\frac{1}{L^a}f(\frac{T}{L^b})$ with $a \neq b$ for large $L$.