Relation between fractal dimension and spatial correlation length for extensive chaos

SUSTAINED nonequiibrium systems can be characterized by a fractal dimension D⩾0, which can be considered to be a measure of the number of independent degrees of freedom 1 . The dimension D is usually estimated from time series 2 but the available algorithms are unreliable and difficult to apply when D is larger than about 5 (refs 3,4). Recent advances in experimental technique 5–8 and in parallel computing have now made possible the study of big systems with large fractal dimensions, raising new questions about what physical properties determine D and whether these physical properties can be used in place of time-series to estimate large fractal dimensions. Numerical simulations 9–11 suggest that sufficiently large homogeneous systems will generally be extensively chaotic 12 , which means that D increases linearly with the system volume V. Here we test an hypothesis that follows from this observation: that the fractal dimension of extensive chaos is determined by the average spatial disorder as measured by the spatial correlation length ε associated with the equal-time two-point correlation function —a measure of the correlations between different regions of the system. We find that the hypothesis fails for a representative spatiotemporal chaotic system. Thus, if there is a length scale that characterizes homogeneous extensive chaos, it is not the characteristic length scale of spatial disorder.