Role of Defects in Self-Organized Criticality: A Directed Coupled Map Lattice Model

TL;DR

This paper investigates how defects influence self-organized criticality in a directed coupled map lattice model, revealing universal critical behavior in conservative dynamics and driven-out-of-criticality in nonconservative cases.

Contribution

It introduces a two-dimensional directed coupled map lattice model with quenched defects, demonstrating universal critical exponents and scaling forms in conservative dynamics and non-universal behavior when energy transfer is incomplete.

Findings

System reaches critical state with universal exponents in conservative case.

Distribution of avalanche durations, sizes, and energies follow scaling laws.

Nonconservative dynamics disrupt criticality, with scaling affected by defect concentration.

Abstract

We study a directed coupled map lattice model in two dimensions, with two degrees of freedom associated with each lattice site. The two freedoms are coupled at a fraction $c$ of lattice bonds acting as quenched random defects. In the case of conservative dynamics, at any concentration of defects the system reaches a self-organized critical state with universal critical exponents close to the mean-field values. The probability distributions follow the general scaling form $P(X,L)= L^{-\alpha}{\cal{P}}(XL^{-D_X})$, where $\alpha \approx 1$ is the scaling exponent for the distribution of avalanche lengths, $X$ stands for duration, size or released energy, and $D_X$ is the fractal dimension with respect to $X$. The distribution of current is nonuniversal, and does not show any apparent scaling form. In the case of nonconservative dynamics---obtained by incomplete energy transfer at the defect bonds--- the system is driven out of the critical state. In the scaling region close to $c=0$ the probability distributions exhibit the general scaling form $P(X,c,L)=X^{-\tau _X }{\cal{P}}(X/\xi _X (c), XL^{-D_X})$, where $\tau _X =\alpha /D_X$ and the coherence length $\xi_X (c)$ depends on the concentration of defect bonds $c$ as $\xi _X (c)\sim c^{-D_X}$.