The Universal Statistical Structure and Scaling Laws of Chaos and Turbulence

Abstract: Turbulence is a complex spatial and temporal structure created by the strong non-linear dynamics of fluid flows at high Reynolds numbers. Despite being an ubiquitous phenomenon that has been studied for centuries, a full understanding of turbulence remained a formidable challenge. Here, we introduce tools from the fields of quantum chaos and Random Matrix Theory (RMT) and present a detailed analysis of image datasets generated from turbulence simulations of incompressible and compressible fluid flows. Focusing on two observables: the data Gram matrix and the single image distribution, we study both the local and global eigenvalue statistics and compare them to classical chaos, uncorrelated noise and natural images. We show that from the RMT perspective, the turbulence Gram matrices lie in the same universality class as quantum chaotic rather than integrable systems, and the data exhibits power-law scalings in the bulk of its eigenvalues which are vastly different from uncorrelated classical chaos, random data, natural images. Interestingly, we find that the single sample distribution only appears as fully RMT chaotic, but deviates from chaos at larger correlation lengths, as well as exhibiting different scaling properties.