RELATION BETWEEN FRACTAL DIMENSION AND SPATIAL CORRELATION LENGTH FOR EXTENSIVE CHAOS

SUSTAINED nonequilibrium systems can be characterized by a fractal dimension D greater than or equal to 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' 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 xi 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.