Abelian deterministic self organized criticality model: Complex dynamics of avalanche waves

TL;DR

This study investigates wave dynamics and size scaling of avalanches in a two-dimensional abelian deterministic self-organized criticality model, revealing multifractal scaling and correlations under various conditions.

Contribution

It extends understanding of avalanche behavior by analyzing correlated waves and multifractal scaling across a range of parameters, challenging the idea that precise relaxation balance is necessary.

Findings

Avalanche size waves are correlated under various conditions.

Multifractal scaling observed in avalanche sizes.

Relaxation balance may be a special case, not a requirement.

Abstract

The aim of this study is to investigate a wave dynamics and size scaling of avalanches which were created by the mathematical model {[}J. \v{C}ern\'ak Phys. Rev. E \textbf{65}, 046141 (2002)]. Numerical simulations were carried out on a two dimensional lattice $L\times L$ in which two constant thresholds $E_{c}^{I}=4$ and $E_{c}^{II}>E_{c}^{I}$ were randomly distributed. A density of sites $c$ with the threshold $E_{c}^{II}$ and threshold $E_{c}^{II}$ are parameters of the model. I have determined autocorrelations of avalanche size waves, Hurst exponents, avalanche structures and avalanche size moments for several densities $c$ and thresholds $E_{c}^{II}$. I found correlated avalanche size waves and multifractal scaling of avalanche sizes not only for specific conditions, densities $c=0.0$, 1.0 and thresholds $8\leq E_{c}^{II}\leq32$, in which relaxation rules were precisely balanced, but also for more general conditions, densities $0.0<c<1.0$ and thresholds $8\leq E_{c}^{II}\leq3 in which relaxation rules were unbalanced. The results suggest that the hypothesis of a precise relaxation balance could be a specific case of a more general rule.