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

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 $\xi _X (c)\sim c^{-D_X}$.