Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium

Abstract

The wave kinetic equation predicts the averaged temporal evolution of a continuous spectral density of waves either randomly interacting or scattered by the fine structure of a medium. In a wide range of systems, the wave kinetic equation is derived from a fundamental equation of wave motion, which is symmetric through time-reversal. By contrast, the corresponding wave kinetic equations is time-irreversible: its solutions monotonically increase an entropy-like quantity. A similar paradox appears whenever one make a mesoscopic description of the evolution of a very large number of microscopic degrees of freedom, the paradigmatic example being the kinetic theory of dilute gas molecules leading to the Boltzmann equation. Since Boltzmann, it has been understood that a probabilistic understanding solves the apparent paradox. More recently, it has been understood that the kinetic description itself, at a mesoscopic level, should not break time reversal symmetry [1]. The time reversal symmetry remains a fundamental property of the mesoscopic stochastic process: without external forcing the path probabilities obey a detailed balance relation with respect to an equilibrium quasipotential. The proper theoretical or mathematical tool to derive fully this mesoscopic time reversal stochastic process is large deviation theory: a This publication is part of a special issue in homage of the memory of Krzysztof Gawȩdzki. The subject of this work is large deviation theory applied to wave turbulence. Large deviation theory applied to complex dynamics and turbulent flows was one of the subjects for which Krzysztof Gawȩdzki made a number of important contributions during the last few years, see for instance [3][4][5][6][7][8]. He taught many of us, including Freddy Bouchet, many aspects of large deviation theory. We wrote a common paper on the subject of large deviation theory and non-equilibrium quasipotentials for stochastic particles with mean field interactions [8]. Given his scientific qualities, and his deep sense of friendship, it is great pleasure for us to pay homage to Krzysztof Gawȩdzki through this modest contribution.