Stochastic wave-kinetic theory in the Liouville approximation

Abstract

The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic, random media is investigated on the basis of a stochastic, collisionless, Liouville‐type equation governing the temporal evolution of a phase‐space Wigner distribution density function. Within the framework of the first‐order smoothing approximation, a general diffusion–convolution‐type kinetic or transport equation is derived for the mean phase‐space distribution function containing generalized (nonloral, with memory) diffusion, friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long‐time, Markovian, diffusion approximation, a general set of Fokker–Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.