Dependence of extensive chaos on the spatial correlation length (substantial revision)

We consider spatiotemporal chaotic systems for which spatial correlation functions decay substantially over a length scale xi (the spatial correlation length) that is small compared to the system size L. Numerical simulations suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L >> xi). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in proportion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos---the Lyapunov dimension density---depends on spatial disorder, as measured by the spatial correlation length xi. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size xi.