Synchronization and Coarsening (without SOC) in a Forest-Fire Model

TL;DR

This paper investigates the long-term behavior of a forest-fire model showing self-organized coarsening into synchronized patches, with analytical and numerical analysis revealing linear growth of patch size and specific scaling laws.

Contribution

It introduces a detailed analysis of patch coarsening without self-organized criticality in a forest-fire model, combining mean-field theory and event-driven simulations.

Findings

Average patch length grows linearly with time.

Patch size distribution follows a specific scaling form.

Numerical results agree with mean-field predictions.

Abstract

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length <L> grows linearly with time as t-->oo. The number density of patches of length L, N(L,t), scales as <L>^{-2}M(L/<L>), and within a mean-field rate equation description we find that this scaling function decays as e^{-1/x} for x-->0, and as e^{-x} for x-->oo. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.