Loop diagrams in the kinetic theory of waves

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 next-to-leading-order 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.