Nonstationary random acoustic and electromagnetic fields as wave diffusion processes

TL;DR

This paper models the stochastic behavior of acoustic and electromagnetic fields in overmoded cavities with dynamic boundaries as a generalized diffusion process, deriving equations and solutions that describe their statistical properties.

Contribution

It introduces a physical model representing field dynamics as a generalized diffusion process and derives related Langevin--Itô and Fokker--Planck equations with solutions.

Findings

Energy diffusion parameter proportional to the square of source field variation

Energy drift vanishes asymptotically

Provides explicit solutions for nonstationary field statistics

Abstract

We investigate the effects of relatively rapid variations of the boundaries of an overmoded cavity on the stochastic properties of its interior acoustic or electromagnetic field. For quasi-static variations, this field can be represented as an ideal incoherent and statistically homogeneous isotropic random scalar or vector field, respectively. A physical model is constructed showing that the field dynamics can be characterized as a generalized diffusion process. The Langevin--It\^{o} and Fokker--Planck equations are derived and their associated statistics and distributions for the complex analytic field, its magnitude and energy density are computed. The energy diffusion parameter is found to be proportional to the square of the ratio of the standard deviation of the source field to the characteristic time constant of the dynamic process, but is independent of the initial energy density, to first order. The energy drift vanishes in the asymptotic limit. The time-energy probability distribution is in general not separable, as a result of nonstationarity. A general solution of the Fokker--Planck equation is obtained in integral form, together with explicit closed-form solutions for several asymptotic cases. The findings extend known results on statistics and distributions of quasi-stationary ideal random fields (pure diffusions), which are retrieved as special cases.