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

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.

EPL-style, 7 pages, 3 eps figures, revised version

References in corpus (2)

• Scaling in self-organized criticality from interface depinning?
• Scaling behavior of the order parameter and its conjugated field in an absorbing phase transition around the upper critical dimension

Cited by in corpus (21)

• Colloquium: Criticality and dynamical scaling in living systems
• Universal scaling behavior of non-equilibrium phase transitions
• Self-organization without conservation: true or just apparent scale-invariance?
• Feedback mechanisms for self-organization to the edge of a phase transition
• Non-ergodicity in the Anisotropic Dicke model
• Driving sandpiles to criticality and beyond
• The approach to criticality in sandpiles
• Self-organized quantization and oscillations on continuous fixed-energy sandpiles
• Exact solution of the one-dimensional deterministic Fixed-Energy Sandpile
• Parallel Chip-Firing on the Complete Graph: Devil's Staircase and Poincare Rotation Number
• Dynamical Invariants in the Deterministic Fixed-Energy Sandpile
• Activity Dependent Branching Ratios in Stocks, Solar X-ray Flux, and the Bak-Tang-Wiesenfeld Sandpile Model
• Asymptotic behavior of the order parameter in a stochastic sandpile
• Connecting the Micro-dynamics to the Emergent Macro-variables: Self-Organized Criticality and Absorbing Phase Transitions in the Deterministic Lattice Gas
• Directed Fixed Energy Sandpile Model
• Abrupt transition in a sandpile model
• Abelian networks IV. Dynamics of nonhalting networks
• On the Period Lengths of the Parallel Chip-Firing Game
• Emergence of oscillations in fixed energy sandpile models on complex networks
• On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator
• The devil's staircase for chip-firing on random graphs and on graphons