Structure and statistical organization of the stationary state of the Oslo model

TL;DR

This paper provides a detailed statistical and structural analysis of the stationary state of the Oslo model, revealing configuration classes and their weights through invariant quantities and diagram counting.

Contribution

It offers the first explicit description of the stationary state of the Oslo model using multiple representations and invariant quantities.

Findings

Configurations form a small number of equivalence classes.

Statistical weights relate to counting colored diagrams with specific rules.

The approach links microscopic rules to macroscopic stationary properties.

Abstract

In most driven-dissipative sandpile models, the dynamics of the system reaches a critical stationary state. This state displays organization features such as a power-law avalanche spectrum and hyperuniformity, but these features often emerge without a clear path from the microscopic evolution rules. Only in a few cases is there an available description of the stationary state, in other sandpile models the question is open. In this article, we present our result on the stationary state of the Oslo model, a driven-dissipative sandpile model with intrinsic randomness. In order to do so, we use different representations of the system configurations and of the dynamical process. Moving back and forth between these representations allows to identify invariant quantities for each configurations. Moreover, we obtain the detailed statistical description of the stationary state by considering all paths leading to a given configuration at once, and by summing their contributions under the constraint specified by the invariants. As a result, we find that the configurations of the stationary state are structured into a small number of equivalence classes, and that their statistical weights are related to the counting of colored diagrams respecting a small set of rules.