The random coupling model of turbulence as a classical SYK model

We point out that a classical analogue of the SYK 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, in which there is mixing between many different modes, the random coupling model has a Gaussian-random coupling between any four of a large $N$ number 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.
16 pages