Fibonacci turbulence

TL;DR

This paper introduces a new family of discrete models with Fibonacci-based conservation laws to analyze turbulence, revealing how cascade directions and non-Gaussian statistics emerge in strongly interacting systems like hydrodynamics and wave interactions.

Contribution

It provides the first detailed information-theoretic analysis of turbulence in systems with Fibonacci conservation laws, uncovering cascade types and statistical behaviors.

Findings

Identified three turbulence types: single direct, double, and inverse cascades.

Quantified how turbulence deviates from thermal equilibrium to non-Gaussian statistics.

Mapped information flow and mode interactions using mutual and interaction information.

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.