Eur. Phys. J. B 52, 91-105 (2006)
https://doi.org/10.1140/epjb/e2006-00262-2

Dynamical invariants in the deterministic fixed-energy sandpile

¹ Dipartimento di Fisica and CNR - INFM, Università di Parma, Parco Area Scienze 7a, 43100 Parma, Italy
² INFN, gruppo collegato di, Parma, Italy
³ Laboratoire de Physique Théorique, Batiment 210, Université de Paris-Sud, 91405 Orsay Cedex, France
⁴ School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

Received: 1 March 2006
Revised: 25 May 2006
Published online: 29 June 2006

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 Laplacians 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 graphs 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.