Calculation of the transition matrix and of the occupation probabilities for the states of the Oslo sandpile model

TL;DR

This paper provides an analytical approach to compute transition probabilities and occupation distributions in the Oslo sandpile model, revealing how system complexity grows with size and enabling exact calculations of avalanche distributions.

Contribution

It introduces an analytic method to determine transition probabilities and stationary distributions in the Oslo sandpile model, advancing understanding of its critical behavior.

Findings

Transition probabilities between configurations are explicitly calculated.

Stationary occupation probabilities vary widely across states.

Exact avalanche size distributions are obtained for given system sizes.

Abstract

The Oslo sandpile model, or if one wants to be precise, ricepile model, is a cellular automaton designed to model experiments on granular piles displaying self-organized criticality. We present an analytic treatment that allows the calculation of the transition probabilities between the different configurations of the system; from here, using the theory of Markov chains, we can obtain the stationary occupation distribution, which tell us that the phase space is occupied with probabilities that vary in many orders of magnitude from one state to another. Our results show how the complexity of this simple model is built as the number of elements increases, and allows, for a given system size, the exact calculation of the avalanche size distribution and other properties related to the profile of the pile.