Loop diagrams in the kinetic theory of waves

TL;DR

This paper advances the kinetic theory of weakly interacting waves by deriving a two-loop kinetic equation and providing a graphical method for higher-order terms, enhancing understanding of wave turbulence.

Contribution

It introduces a systematic graphical approach to compute the kinetic equation at any order, extending previous perturbative methods in wave turbulence theory.

Findings

Derived the two-loop kinetic equation for cubic wave interactions

Provided a graphical prescription for higher-order kinetic terms

Enhanced the theoretical framework for weak wave turbulence

Abstract

Recent work has given a systematic way for studying the kinetics of classical weakly interacting waves beyond leading order, having analogies with renormalization in quantum field theory. An important context is weak wave turbulence, occurring for waves which are small in magnitude and weakly interacting, such as those on the surface of the ocean. Here we continue the work of perturbatively computing correlation functions and the kinetic equation in this far-from-equilibrium state. In particular, we obtain the two-loop kinetic equation for waves with a cubic interaction. Our main result is a simple graphical prescription for the terms in the kinetic equation, at any order in the nonlinearity.