Transport and diffusion in a statistical medium

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Abstract
The problem of formulating and solving linear kinetic (transport) equations in a random maedium is considered. In such a medium, the cross sections and external source are known only in a statistical sense. A projection operator technique is used to derive the appropriate equation for the ensemble averaged distribution function. This equation contains and infinite series, with the nth term involving an n-fold applicaytion of the inverse operator for a non-statistical transport equation. This multiple inverse operator (a multiple integral over Green's functions) acts on various correlation functions describing the statistical nature of the medium. Fokker-Planck approximation can be employed to localize these integral operators. For a static statistical mixture of two fluids, a stationary Markov model is assumed to compute the required correlations. Various explicit solutions to the ensemble averaged transport equation, both exact and approximate, are given for the purely absorbing (no scattering) time independent case, with and without external sources. The accuracy of truncating the infinite series and using the Fokker-Planck approximation is discussed, and sample numerical results given. An independent approach and solution, involving the distribution function for the optical depth, is applied to the source free probĺem, and this exact result is shown to agree with that obtained by the projection operator technique (with no truncation and no Fokker-Planck approximation). Finally, a simple procedure is suggested for “homogenizing” the statistical two fluid medium. This yields an effective collision cross section and external source which accounts for the statistical nature of the problem.

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