Kinetic theory of waves in random media and amelioration of classical chaos.

Abstract

The approach to the classical limit of wave mechanics is investigated where, in the classical limit, the dynamical system is nonintegrable and the motion in phase space is chaotic. The problem is cast in the setting of wave propagation in random media, and the fundamental starting point is an idealized stochastic parabolic wave equation (SPE) in two space dimensions with plane wave initial data. The potential is taken to have mean zero, strength ε ≪ 1 fluctuations which are homogeneous, isotropic, and have a single scale. The formal classical limit of the SPE, the parabolic ray equations are inherently non-integrable for any given realization of the potential. For the relative motion of two particles, an advection-diffusion Fokker-Planck equation is derived and shown for small initial separations to exhibit chaotic behavior, characterized by the existence of a positive Lyapunov exponent. It is shown that this physically relates to the exponential proliferation of caustics, or tendrils in phase space. A generalized wave kinetic equation (GWKE) is derived for the evolution in a relative phase space of a mean, two-particle Wigner function which corresponds classically to the advection-diffusion Fokker-Planck equation. The GWKE is analytically examined semi-classically by a novel boundary layer method (called the "extended quantum notch method") which enable the derivation of several important results: First, the "log time" (range) is obtained where semi-classical theory breaks down due to the saturation of caustics, then it is shown that this range is where the normalized intensity fluctuations (scintillation index) approaches unity. Finally, a wave (quantum) manifestation of classical chaos is seen to be the exponential decay of the scintillation index beyond its peak while on approach to saturation.