We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.

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Benchmark results for particle transport in a binary Markov statistical medium

A statistical analysis of the double heterogeneity problem

For time-independent transport in planar systems, a formalism is developed to treat radiative transfer in a statistical medium, including the scattering interaction, for a certain class of problems. In this development, the statistics of the mixture are allowed to be completely general. As a special case, the purely scattering rod problem described by homogeneous Markov statistics is treated in detail, and numerical results given. These results agree with those reported earlier based upon the method of invariant imbedding. In particular, the existence of a transmission window is confirmed for this problem.

Radiative transfer in random media with scattering

A model involving integral renewal equations is derived which describes the ensemble-averaged intensity for time-independent particle transport in a nonscattering medium composed of two randomly mixed, immiscible fluids. It is shown that for a special class of mixing statistics these renewal equations can be recast into integro-differential equations resembling usual transport/kinetic equations. An extension of this model is proposed which includes time dependence as well as scattering. This generalizes an earlier description proposed for the special case of Markov mixing statistics. An Appendix is included which gives several statistical results useful in this line of research.

Statistics, renewal theory, and particle transport

Linear kinetic particle transport in stochastic heterogeneous media is discussed. The analysis includes scattering in a three‐dimensional setting and deals with arbitrary time‐dependent statistics. Ensemble‐average operators are used to derive two independent complete descriptions for the ensemble‐averaged angular flux. The first description consists of an infinite system of integral, renewal‐like equations for averaged flux values over spatially dependent, increasingly smaller sets. The second approach results in an infinite system of kinetic, balancelike equations for locally averaged flux values. Both types of equations include averagings over transitional sets of states that change locally of physical properties. Previous results are recovered from the limit form of these equations for no‐memory statistics and purely absorbing media. Also, the Levermore–Pomraning–Wong proposed models are shown to correspond to truncated forms of these equations.

Linear kinetic theory in stochastic media

As a generalization of Markov processes, renewal processes are proposed to describe any statistically homogeneous and isotropic, two-component, random medium with structure. An effective opacity is calculated in the diffusion limit as a function of the chord-length distributions through the grains of matter.

D. Vanderhaegen

Papers

Impact of a mixing structure on radiative transfer in random media