Exact solution of the one-dimensional deterministic Fixed-Energy Sandpile

TL;DR

This paper provides an exact solution to the one-dimensional deterministic Fixed-Energy Sandpile model, revealing the origins of its non-ergodic behavior and characterizing its phase diagram based on energy density.

Contribution

It offers the first exact analytical solution for the model, elucidating the structure of periodic orbits and their dependence on energy, advancing understanding of non-ergodicity in such systems.

Findings

Existence of well-defined periodic orbits depending on energy density

Complete characterization of the activity vs. energy phase diagram

Analytical insight into the non-ergodic dynamics of the model

Abstract

In reason of the strongly non-ergodic dynamical behavior, universality properties of deterministic Fixed-Energy Sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the non-ergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction's basins of the different periodic orbits lead to a complete characterization of the activity vs. energy density phase diagram in the limit of large system's size.