Defects in self-organized criticality: A directed coupled map lattice model

We study a directed coupled map lattice model in d=2 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. The system is driven (by adding ‘‘energy,’’ say) in one of the degrees of freedom at the top of the lattice, and the relaxation rules depend on the local difference between the two variables at a lattice site. 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 ${\mathrm{\tau }}_{\mathit{t}}$=1, ${\mathrm{\tau }}_{\mathit{s}}$=2/3, and ${\mathrm{\tau }}_{\mathit{n}}$=1/2, for the integrated distributions of avalanche durations (t), size (s), and released energy (n), respectively. The probability distributions follow the general scaling form P(X,L)=${\mathit{L}}^{\mathrm{-}\mathrm{\alpha }}$ P(${\mathit{XL}}^{\mathrm{-}{\mathit{D}}_{\mathit{X}}}$), where α≊1 is the scaling exponent for the distribution of avalanche lengths, X stands for t, s, or n, and ${\mathit{D}}_{\mathit{X}}$ is the (independently determined) fractal dimension with respect to X. The distribution of current through the system is, however, 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)=${\mathit{X}}^{\mathrm{-}{\mathrm{\tau }}_{\mathit{X}}}$ P[X/${\xi }_{\mathit{X}}$(c),${\mathit{XL}}^{\mathrm{-}{\mathit{D}}_{}\mathrm{X}}$], where ${\mathrm{\tau }}_{\mathit{X}}$=α/${\mathit{D}}_{\mathit{X}}$ and the corresponding coherence length ${\xi }_{\mathit{X}}$(c) depends on the concentration of defect bonds c as ${\xi }_{\mathit{X}}$(c)∼${\mathit{c}}^{\mathrm{-}{\mathit{D}}_{\mathit{X}}}$. © 1996 The American Physical Society.