Abstract
Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first case of a detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with neighboring triplet interactions and show that it has families of quadratic conservation laws defined by the Fibonacci numbers. Depending on the single model parameter, three types of turbulence were found: single direct cascade, double cascade, and the first ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, as quantified by the mutual information in pairs and the interaction information inside triplets.
Highlights
The existence of quadratic invariants and Gaussianity of equilibrium in a strongly interacting system may seem exceptional
Traditional study of turbulence in general and shell models in particular was focused on the single-mode moments, hjaijqi ∝ F−i ζq, whose anomalous scaling exponents ΔðqÞ 1⁄4 qζ3=3 − ζq give particular measures of how nonGaussianity grows along the cascade
We find that the neighboring multipliers are dependent, albeit weakly, as expressed in their mutual information
Summary
The existence of quadratic invariants and Gaussianity of equilibrium in a strongly interacting system may seem exceptional. Two very distinct wide classes of physical systems have quadratic invariants and Gaussian statistics at thermal equilibrium. The statistical physics approach to turbulence was to a large extent devoted to two quite distinct classes: systems of interacting waves like those on the surface of the ocean or a puddle and incompressible vortical flows where no waves are possible. We consider the particular subclass of models that allow only neighboring interactions, and find it the most versatile tool to date to study turbulence as an ultimate far-from-equilibrium state. We carry here such detailed study of the known types of direct-only and double cascades with unprecedented numerical resolution. Our models allow for an inverse-only cascade never encountered before
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