Probability distribution of residence times of grains in models of ricepiles

TL;DR

This paper analyzes the probability distribution of grain residence times in ricepile models, revealing decay behaviors and scaling properties, with implications for understanding deep burial effects and model dimensionality.

Contribution

It introduces a detailed analysis of residence time distributions in ricepile models, highlighting non-monotonic behaviors and dimensional scaling laws.

Findings

Residence time distributions decay as 1/t(ln t)^x for large t.

Probability of minimum slope configuration scales as exp(-κL^{d+2}).

Residence times at a site can be non-monotonic with system size.

Abstract

We study the probability distribution of residence time of a grain at a site, and its total residence time inside a pile, in different ricepile models. The tails of these distributions are dominated by the grains that get deeply buried in the pile. We show that, for a pile of size $L$, the probabilities that the residence time at a site or the total residence time is greater than $t$, both decay as $1/t(\ln t)^x$ for $L^{\omega} \ll t \ll \exp(L^{\gamma})$ where $\gamma$ is an exponent $ \ge 1$, and values of $x$ and $\omega$ in the two cases are different. In the Oslo ricepile model we find that the probability that the residence time $T_i$ at a site $i$ being greater than or equal to $t$, is a non-monotonic function of $L$ for a fixed $t$ and does not obey simple scaling. For model in $d$ dimensions, we show that the probability of minimum slope configuration in the steady state, for large $L$, varies as $\exp(-\kappa L^{d+2})$ where $\kappa$ is a constant, and hence $ \gamma = d+2$.