Self-organized criticality as an absorbing-state phase transition

TLDR

Self‑organized criticality in sandpile models arises because the system lacks energy saturation and is defined at the critical point (ε=0, h=0+), a feature shared with other absorbing‑state transitions. The study investigates how self‑organized criticality relates to absorbing‑state phase transitions, argues that power‑law avalanches emerge generically under slow driving at criticality, and develops a field theory coupling the order parameter to conserved energy. The authors employ a field‑theoretic framework that couples the order parameter to a conserved energy density and analyze systems under slow driving at the critical point. Simulations demonstrate that sandpile models become critical only when both dissipation ε and driving field h are zero, confirming that power‑law avalanches arise at the critical point under slow driving.

Abstract

We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters --- dissipation $\ensuremath{\epsilon}$ and driving field $h$ --- are set to their critical values. The critical values of $\ensuremath{\epsilon}$ and $h$ are both equal to zero. The first result is due to the absence of saturation (no bound on energy) in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point $(\ensuremath{\epsilon}{=0,h=0}^{+}):$ it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at $\ensuremath{\epsilon}=h=0$ and fixed energy density $\ensuremath{\zeta}$ (no drive, periodic boundaries), and of the slowly driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate.