Path large deviations for the kinetic theory of weak turbulence

Abstract

We consider a generic Hamiltonian system of nonlinear interacting waves with 3-wave interactions. In the kinetic regime of wave turbulence, which assumes weak nonlinearity and large system size, the relevant observable associated with the wave amplitude is the empirical spectral density that appears as the natural precursor of the spectral density, or spectrum, for finite system size. Following classical derivations of the Peierls equation for the moment generating function of the wave amplitudes in the kinetic regime, we propose a large deviation estimate for the dynamics of the empirical spectral density, where the number of admissible wavenumbers, which is proportional to the volume of the system, appears as the natural large deviation parameter. The large deviation stochastic Hamiltonian that quantifies the minus of the log-probability of a trajectory is computed within the kinetic regime which assumes the Random Phase approximation for weak nonlinearity. We compare this Hamiltonian with the one for a system of modes interacting in a mean-field way with the empirical spectrum. Its relationship with the Random Phase and Amplitude approximation is discussed. Moreover, for the specific case when no forces and dissipation are present, a few fundamental properties of the large deviation dynamics are investigated. We show that the latter conserves total energy and momentum, as expected for a 3-wave interacting systems. In addition, we compute the equilibrium quasipotential and check that global detailed balance is satisfied at the large deviation level. Finally, we discuss briefly some physical applications of the theory.

Rigorous justification of the wave kinetic theory

“…Heuristically, it can be stated as the requirement that the minimum separation between values of ω(k) with nearby k is much less than the width δ of the window W δ . This leads to the condition [39,33] 1…”

Derivation of the wave kinetic equation: full range of scaling laws

“…Clearly, the range of admissibility depends on the precise properties of the dispersion relation ω. Note that a sufficient condition is given by 1 L ∂ω ∂k α 2 which corresponds to γ ≤ 1/2 [36,27]: in this range, the equidistribution property holds for any reasonably behaved dispersion relation ω without the need for any number theoretic arguments. However, for a given (or a class of) dispersion relation ω, the above sufficient condition is usually not necessary.…”