Probability distribution of residence times of grains in models of rice piles

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 rice pile 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) << t << exp(L(gamma)) where gamma is an exponent > or = 1, and values of x and omega in the two cases are different. In the Oslo rice pile model we find that the probability of the residence time T(i) at a site i being greater than or equal to t is a nonmonotonic 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(-kappaL(d+2)) where kappa is a constant, and hence gamma=d+2.