Feynman rules for forced wave turbulence

TL;DR

This paper develops Feynman diagram techniques to compute correlation functions in forced wave turbulence, extending the understanding of fluctuations around non-equilibrium turbulent states in classical field theories.

Contribution

It introduces a Feynman diagram approach for classical stochastic field theories with quartic interactions, enabling calculation of correlation functions beyond leading order.

Findings

Derived the kinetic equation to next-to-leading order.

Computed two-point and four-point functions explicitly.

Connected classical turbulence with quantum field theory methods.

Abstract

It has long been known that weakly nonlinear field theories can have a late-time stationary state that is not the thermal state, but a wave turbulent state with a far-from-equilibrium cascade of energy. We go beyond the existence of the wave turbulent state, studying fluctuations about the wave turbulent state. Specifically, we take a classical field theory with an arbitrary quartic interaction and add dissipation and Gaussian-random forcing. Employing the path integral relation between stochastic classical field theories and quantum field theories, we give a prescription, in terms of Feynman diagrams, for computing correlation functions in this system. We explicitly compute the two-point and four-point functions of the field to next-to-leading order in the coupling. Through an appropriate choice of forcing and dissipation, these correspond to correlation functions in the wave turbulent state. In particular, we derive the kinetic equation to next-to-leading order.