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 freedom1. The dimension D is usually estimated from time series2 but the available algorithms are unreliable and difficult to apply when D is larger than about 5 (refs 3,4). Recent advances in experimental technique5-8 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 simulations9-11 suggest that sufficiently large homogeneous systems will generally be extensively chaotic12, 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. © 1994 Nature Publishing Group.

Duke Scholars

Author Henry Greenside Physics