Spatiotemporal Chaos in Large Systems: The Scaling of Complexity with Size

Abstract

The dynamics of a nonequilibrium system can become complex because the system has many components (e.g., a human brain), because the system is strongly driven from equilibrium (e.g., large Reynolds-number flows), or because the system becomes large compared to certain intrinsic length scales. Recent experimental and theoretical work is reviewed that addresses this last route to complexity. In the idealized case of a sufficiently large, nontransient, homogeneous, and chaotic system, the fractal dimension D becomes proportional to the system's volume V which defines the regime of extensive chaos. The extensivity of the fractal dimension suggests a new way to characterize correlations in high-dimensional systems in terms of an intensive dimension correlation length $ξ_δ$. Recent calculations at Duke University show that $ξ_δ$ is a length scale smaller than and independent of some commonly used measures of disorder such as the two-point and mutual-information correlation lengths. Identifying the basic length and time scales of extensive chaos remains a central problem whose solution will aid the theoretical and experimental understanding of large nonequilibrium systems.