Feynman rules for forced wave turbulence

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.

A chaotic lattice field theory in one dimension*

“…We work 'beyond perturbation theory', in the anti-integrable, strong coupling regime, in contrast to much of the literature that focuses on weak coupling expansions around a 'ground state'. And, in contrast to [11,92,128,157,183], our 'far from equilibrium' field theory has no added dissipation, and is not driven by external noise. All chaoticity is due to the intrinsic unstable deterministic dynamics, and our trace formulas (11) are exact, not merely saddle points approximations to the exact theory.…”