Fokker-Planck approach to wave turbulence

TL;DR

This paper reexamines wave turbulence stationary states using a Fokker-Planck framework, introducing a non-perturbative approach and analyzing the limitations of thermal equilibrium within this context.

Contribution

It presents a novel Fokker-Planck approach to wave turbulence, connecting kinetic equations with quantum Ehrenfest-like relations and addressing equilibrium state issues.

Findings

Reformulation of wave turbulence kinetic equations via Fokker-Planck Hamiltonian.

Identification that thermal equilibrium is not a stationary state in this framework.

Proposal of a nonlinear dissipation modification to achieve proper equilibrium.

Abstract

The Kolmogorov-Zakharov stationary states for weak wave turbulence involve solving a leading-order kinetic equation. Recent calculations of higher-order corrections to this kinetic equation using the Martin-Siggia-Rose path integral are reconsidered in terms of stationary states of a Fokker-Planck Hamiltonian. A non-perturbative relation closely related to the quantum mechanical Ehrenfest theorem is introduced and used to express the kinetic equation in terms of divergences of two-point expectation values in the limit of zero dissipation. Similar equations are associated to divergences in higher-order cumulants. It is additionally shown that the ordinary thermal equilibrium state is not actually a stationary state of the Fokker-Planck Hamiltonian, and a non-linear modification of dissipation is considered to remedy this.