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 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.
- Received 27 January 2021
- Revised 31 March 2021
- Accepted 29 April 2021
DOI:https://doi.org/10.1103/PhysRevX.11.021063
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Studies of turbulence encompass a wide variety of phenomena in nature and industry, from pipe flows to ripples on a puddle. So far, the statistical physics approach to turbulence has been largely devoted to two distinct classes: systems of interacting waves like those on the surface of the ocean or an incompressible flow where no waves are possible. Here, we build a bridge between these two classes and show that a certain kind of discrete model can describe both. In these systems, the difference between thermal equilibrium and turbulence is dramatic: In thermal equilibrium, physical variables at different length scales fluctuate independently, whereas turbulence is characterized by intricate correlations. We explore this distinction and introduce a way to disentangle correlations.
A turbulent steady state of a system can be reached when energy is constantly injected into the system and dissipated at very distinct length scales in a balanced way. A quantity that is conserved in a closed system usually “cascades” through different scales in the open system; that is, it is transferred from the scale where energy is injected to the scale where energy is dissipated.
We introduce a beautifully simple yet rich family of discrete models and observe three types of turbulence: single direct cascade, where energy transfers from large to small scales; the first-ever case of a single inverse cascade, where energy transfers from small to large scales without an accompanying direct cascade for another conserved quantity; and a double cascade, where direct and inverse cascades exist simultaneously.
Our analysis elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it.
