A CLASS OF TRANSPORT PROBLEMS IN STATISTICAL MIXTURES WITH SCATTERING

Abstract

For time independent transport in one dimensional systems, a formalism is discussed to treat radiative transfer in statistical mixtures, including the scattering interaction, for a certain class of problems. As a special case, binary homogeneous Markov statistics are treated in detail, for both the model of transport and planar geometry. Results are given for several classical transport problems. In the case of the Milne problem, an ambiguity in its definition is pointed out, and the solution is discussed for two different interpretations. 1 INTRODUCTION In recent years there has been considerable interest in developing a formalism to describe radiative transfer (or any linear transport) in a statistical medium consisting of two or more immiscible fluids /1-9/. A complete and rigorous description has been given for time independent problems in the absence of the scattering interaction in the underlying physics of the transport process, both for Markov statistics /1,2/ and non-Markov statistics /3-5/. With scattering and/or time dependence present, a general phenomenological model has been suggested /4,5/ for Markov statistics, using the master equation approach /10/. This model gives the correct result for time independent problems with no scattering, and is always a robust one, but it has recently been shown /6,7/ to give an inexact result for a certain time independent purely scattering problem. This led to the development of a formalism to treat a certain restricted class of statistical time independent transport problems including the scattering interaction /7,8/. In this paper we review some explicit results for certain classic transport problems obtained with this formalism /7-9/. Specifically, we give the results for a transmission problem, the halfspace albedo and emissivity problems, and the Milne problem. In all cases, the statistics of the two fluid random mixture are assumed homogeneous and Markovian. Results are given for both the rod model /ll/ of transport, in which particles are constrained to move along a line, and in the more physical planar geometry. A particularly interesting point is that, in this statistical setting, there is a nonuniqueness in the definition of the Milne problem, which leads to two different approaches giving two different results /9/. 2 GENERAL FORMALISM The starting point of our analysis is the monoenergetic, time independent linear transport equation in planar geometry, given by U) Partially supported by the U.S. National Science Foundation Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988739 C7-322 JOURNAL DE PHYSIQUE Here I(x,/i) is the radiative, or particle, intensity defined as the product of the particle speed and the particle number distribution function, with x and n representing the spatial and angular variables respectively. Specifically, ft is the cosine of the angle between the particle velocity and the positive x axis. The quantity <r(x) is the total (collision) cross section, and os(x,/i'-*/*) denotes the scattering kernel describing the scattering of particles from angle cosine p' to angle cosine /». Q(x,/i) represents any external source of particles. As indicated, Eq. (1) holds for a planar system occupying 0 < x < t. The quantities a, as, and Q in Eq. (1) are treated as stochastic variables, each obeying known statistics. Nonstochastic boundary conditions are assigned to Eq. (1) of the form (2) (3) where r+(/i) are the known boundary data. We seek a solution for I(x,p), the ensemble averaged solution of Eqs. (l)-(3). The key to the analysis is the introduction of an optical depth variable r defined as