Dynamical Large Deviations for an Inhomogeneous Wave Kinetic Theory: Linear Wave Scattering by a Random Medium

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 makes 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 (Bouchet in J Stat Phys 181(2):515-550, 2020). 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 large deviation principle uncovers a time reversible field theory, characterized by a large deviation Hamiltonian, for which the deterministic wave kinetic equation appears as the most probable evolution. Its irreversibility appears as a consequence of an incomplete description, rather than as a consequence of the kinetic limit itself, or some related chaotic hypothesis. This paper follows Bouchet (J Stat Phys 181(2):515-550, 2020) and a series of other works that derive the large deviation Hamiltonians of the main classical kinetic theories, for instance, Guioth et al. (J Stat Phys 189(20):20, 2022) for homogeneous wave kinetics. We propose here a derivation of the large deviation principle in an inhomogeneous situation, for the linear scattering of waves by a weak random potential. This problem involves microscopic scales corresponding to the typical wavelengths and periods of the waves and mesoscopic ones which are the scales of spatial inhomogeneities in the spectral density of both the random scatterers and the wave spectrum, and the time needed for the random scatterers to alter the wave spectrum. The main assumption of the kinetic regime is a large separation of these microscopic and mesoscopic scales. For the sake of simplicity, we consider a generic model of wave scattering by weak disorder: the Schrödinger equation with a random potential. We derive the path large deviation principle for the local spectral density and discuss its main properties. We show that the mesoscopic process obeys a time reversal symmetry at the level of large deviations.