Dynamical Invariants in the Deterministic Fixed-Energy Sandpile

TL;DR

This paper characterizes dynamical invariants in the deterministic Fixed Energy Sandpile, explaining non-ergodic behavior and revealing a complex structure of invariant sets and phase transitions.

Contribution

It provides an explicit algorithm for identifying all toppling invariants and analyzes the configuration space structure, highlighting differences from other sandpile models.

Findings

Number of invariant sets equals graph complexity

Exponential growth of invariant sets with lattice size

Transition in the frozen phase related to energy constraints

Abstract

The non-ergodic behavior of the deterministic Fixed Energy Sandpile (DFES), with Bak-Tang-Wiesenfeld (BTW) rule, is explained by the complete characterization of a class of dynamical invariants (or toppling invariants). The link between such constants of motion and the discrete Laplacian's properties on graphs is algebraically and numerically clarified. In particular, it is possible to build up an explicit algorithm determining the complete set of independent toppling invariants. The partition of the configuration space into dynamically invariant sets, and the further refinement of such a partition into basins of attraction for orbits, are also studied. The total number of invariant sets equals the graph's complexity. In the case of two dimensional lattices, it is possible to estimate a very regular exponential growth of this number vs. the size. Looking at other features, the toppling invariants exhibit a highly irregular behavior. The usual constraint on the energy positiveness introduces a transition in the frozen phase. In correspondence to this transition, a dynamical crossover related to the halting times is observed. The analysis of the configuration space shows that the DFES has a different structure with respect to dissipative BTW and stochastic sandpiles models, supporting the conjecture that it lies in a distinct class of universality.