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

We present a model for steady state winds of systems with super-Eddington luminosities. These radiatively driven winds are expected to be optically thick and clumpy as they arise from an instability driven porous atmosphere. The model is then applied to derive the mass loss observed in bright classical novae. The main results are: (i) A general relation between the mass loss rate and the total luminosity in superEddington systems. (ii) A natural explanation to the long duration super-Eddington outflows that are clearly observed in at least two cases (Novae LMC 1988 #1 & FH Serpentis). (iii)A quantitative agreement between the observed luminosity evolution which is used to predict both the mass loss and temperature evolution, and their observations. (iv)An agreement between the predicted average integrated mass loss of novae as a function of WD mass and its observations. (v) A natural explanation for the ‘transition phase’ of novae. (vi)Agreement with η Carinae which was used to double check the theory. The prediction for the mass shed in the star’s great eruption agrees with observations to within the measurement error.

The theory of steady‐state super‐Eddington winds and its application to novae

A statistical analysis of the double heterogeneity problem

This article is devoted to the mathematical analysis of various formulas giving the equivalent absorption and scattering cross section for mixed materials in linear transport theory. We begin with a general result on the treatment of high-frequency oscillations in linear transport equations which is partly based upon the velocity averaging results and is the analogue, for transport equations, of the compensated compactness class of results. The case of periodic inhomogeneities is then studied in detail; in particular we show the essential difference with periodic homogenization of diffusion equations, due to small divisor problems. These results were announced in [F. Golse, C.R. Acad. Sci. Paris Ser. I Math., 305 (1987), pp. 801--804; F. Golse, Mathematical Aspects of Fluid and Plasma Dynamics, G. Toscani, V. Boffi, and S. Rionero, eds., Lecture Notes in Math 1460, Springer-Verlag, Berlin, New York, 1991, pp. 152--169]. Finally, we treat a case of stochastic inhomogeneities in linear transport theory insp...

Homogenization of transport equations

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

Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model, as well as the simpler model involving the effective cross sections and source.

C. D. Levermore

Papers

Renewal theory for transport processes in binary statistical mixtures