Strong wave turbulence in strongly local large $N$ theories

TL;DR

This paper investigates wave turbulence in large N theories with strongly local interactions, deriving a unified kinetic equation applicable at all coupling strengths and identifying stationary solutions with various scaling behaviors.

Contribution

It derives a kinetic equation valid for all interaction strengths in large N theories with local interactions, and finds stationary solutions exhibiting different turbulence spectra.

Findings

At weak coupling, solutions match Kolmogorov-Zakharov scaling.

At strong coupling, solutions approach generalized Phillips or Kolmogorov-like spectra.

Kinetic equation simplifies to a differential form due to locality in momentum space.

Abstract

We study wave turbulence in systems with two special properties: a large number of fields (large $N$) and a nonlinear interaction that is strongly local in momentum space. The first property allows us to find the kinetic equation at all interaction strengths -- both weak and strong, at leading order in $1/N$. The second allows us to turn the kinetic equation -- an integral equation -- into a differential equation. We find stationary solutions for the occupation number as a function of wave number, valid at all scales. As expected, on the weak coupling end the solutions asymptote to Kolmogorov-Zakharov scaling. On the strong coupling end, they asymptote to either the widely conjectured generalized Phillips spectrum (also known as critical balance), or a Kolmogorov-like scaling exponent.