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Fibonacci Turbulence
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DOI：10.1103/PhysRevX.11.021063.png)
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Abstract
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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 detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis involves both energy and entropy and 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 triplet interactions of neighboring modes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, 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.
Keywords:
SHELL MODELS
CASCADE
FLUID
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