Time transport correlations in abelian sandpile models
Valentin Lallemant
TL;DR
The paper addresses time transport correlations in abelian sandpile models, a challenging problem due to their out-of-equilibrium, nonlinear dynamics. It develops a rigorous framework that leverages abelianity to decompose super avalanches into smaller ones, derives a recursion for the second moment of transported quantity, and proves bounded central moments for integrated observables. By coupling these results to a linear system that relates time correlations to the variance of integrated transport, the authors connect dynamic and static transport properties and obtain an exact expression for time correlations in the Directed Stochastic Sandpile, linking $\langle sW^t s\rangle$ to $\langle s^2(t)\rangle$. The approach applies to several 1D models, including Oslo and Activated Random Walk under suitable conditions, and reveals a quasi integrable structure with conserved statistical quantities at long times, suggesting avenues for extending to density correlations in broader classes of abelian SOC systems.
Abstract
Sandpiles form one of the largest class of models displaying a critical stationary state. Despite a few decades of research, a comprehensive and systematic rigorous characterisation of their spatial and, even more, time dependent properties has remained elusive. Among the obstacles, we can mention their out of equilibrium and non-linear dynamics features which prevent, in general, the access to the stationary properties explicitly. In fact, even the knowledge of the stationary state is quite exceptional in sandpiles. In that respect, it has become standard to develop a model to model strategy and, so to say, general results or tools applicable to these systems are missing. In this paper, we unveil general and simple properties of time transport correlations in certain classes of abelian sandpile models. We proceed gradually, starting from results applicable in a broad context, to more and more specific ones, consequently valid to smaller and smaller classes. For instance, we show, under a few hypothesis, that the number of particles dissipated displays mostly anticorrelation in time. Besides, on a more integrable point of view, the approach followed might culminate with the proof of a link between 2-points time transport correlations and the second moment of the integrated transport over time. To be clear, these two quantities are related through a linear system of equations which is explicitly solved and applies to at least three 1D sandpile models, namely the Directed Stochastic Sandpile, the Oslo and the Activated Random Walk (in a peculiar setup) models.