Short period attractors and non-ergodic behavior in the deterministic fixed energy sandpile model

TL;DR

This paper investigates the fixed energy sandpile model, revealing non-ergodic limit cycles and a devil's staircase pattern in activity versus energy, driven by the model's symmetries.

Contribution

It uncovers the existence of non-ergodic limit cycles and the devil's staircase behavior in the fixed energy sandpile model, linking these phenomena to underlying symmetries.

Findings

Presence of non-ergodic limit cycles depending on energy and initial conditions.

Devil's staircase pattern in activity versus energy density.

Symmetry-based explanation of phase diagram features.

Abstract

We study the asymptotic behaviour of the Bak, Tang, Wiesenfeld sandpile automata as a closed system with fixed energy. We explore the full range of energies characterizing the active phase. The model exhibits strong non-ergodic features by settling into limit-cycles whose period depends on the energy and initial conditions. The asymptotic activity $\rho_a$ (topplings density) shows, as a function of energy density $\zeta$, a devil's staircase behaviour defining a symmetric energy interval-set over which also the period lengths remain constant. The properties of $\zeta$-$\rho_a$ phase diagram can be traced back to the basic symmetries underlying the model's dynamics.