Transport in discrete stochastic mixtures

Abstract

We consider steady-state particle transport through a stochastic mixture of two (or more) immiscible materials. Certain exact results, as well as approximate models, are described for ⟨ψ⟩, the ensemble average of the particle angular flux. In the simplest case of transport in a no-scattering medium, an exact description for ⟨ψ⟩ and higher stochastic moments is easily constructed. If the material mixing is taken as Markovian, the Liouville master equation leads to two (for a binary mixture) coupled deterministic differential transport equations describing ⟨ψ⟩, with similar results for higher stochastic moments, such as the variance. For non-Markovian mixing, renewal analysis leads to four coupled deterministic integral equations describing ⟨ψ⟩ and higher moments. In the more general problem involving scattering, no exact results (except in very special cases) are available, but reasonably accurate models can be developed. Certain asymptotic limits of these models lead to a single, renormalized, transport equation for ⟨ψ⟩ of the standard form, with effective values of the cross sections and source which account for the stochasticity of the problem. Diffusive approximations also result from asymptotic analyses.