Random coupling model of turbulence as a classical Sachdev-Ye-Kitaev model

TL;DR

This paper reveals that a classical turbulence model with random mode couplings, studied in the 1960s, shares mathematical similarities with the quantum Sachdev-Ye-Kitaev (SYK) model, offering new insights into turbulence and quantum chaos.

Contribution

It establishes a connection between the classical turbulence random coupling model and the SYK model using path integral techniques, providing a new physical context for SYK and turbulence studies.

Findings

Derived the effective action for the classical turbulence model.

Showed the large-N saddle point leads to an integral equation similar to SYK.

Suggested new approaches for studying turbulence and quantum chaos.

Abstract

We point out that a classical analog of the Sachdev-Ye-Kitaev model -- a solvable model of quantum many-body chaos, was studied long ago in the turbulence literature. Motivated by the Navier-Stokes equation in the turbulent regime and the nonlinear Schr\"odinger equation describing plasma turbulence, in which there is mixing between many different modes, the random coupling model has a Gaussian-random coupling between any four of a large number $N$ of modes. The model was solved in the 1960s, before the introduction of large $N$ path integral techniques, using a method referred to as the direct interaction approximation. We use the path integral to derive the effective action for the model. The large-$N$ saddle gives an integral equation for the two-point function, which is very similar to the corresponding equation in the SYK model. The connection between the SYK model and the random coupling model may, on the one hand, provide new physical contexts in which to realize the SYK model and, on the other hand, suggest new models of turbulence and techniques for studying them.