Algorithms for Monte Carlo particle transport in binary statistical mixtures

There has been considerable interest lately in particle transport through randomly mixed materials. We focus here on two-material mixtures. Given a particular medium, we do not know the precise composition; we know only a statistical description. We assume here Markovian mixtures, which means chord lengths through each material satisfy an exponential probability distribution. We seek the ensemble-averaged'' transport solution which, given infinite resources, we could compute as follows: construct a particular realization'' of the medium by sampling from the statistical descriptions, solve the transport equation for that realization, and repeat many times and average the solutions. This exact'' procedure is far too costly for practical use; thus, we need efficient algorithms that produce reasonable approximations to the exact solution. Many researchers have sought a set of equations that approximately describe the ensemble-average solution. One such set, which we call the Levermore equations'', has received considerable attention. McCormick has avoided deriving equations, proposing instead a Monte Carlo algorithm for an approximate ensemble-average solution. Pomraning subsequently argued that this algorithm actually solves the Levermore equations. Zimmerman independently proposed a Monte Carlo algorithm that should also solve the Levermore equations, and a second algorithm that should be more accurate. In this work wemore » study three Monte Carlo algorithms: algorithm A, algorithm B, and algorithm C. 8 refs., 1 fig.« less