On the Accuracy of the Non-Classical Transport Equation in 1-D Random Periodic Media

We present a first numerical investigation of the accuracy of the recently proposed {\em non-classical transport equation}. This equation contains an extra independent variable (the path-length $s$), and models particle transport taking place in random media in which a particle's distance-to-collision is {\em not} exponentially distributed. To solve the non-classical equation, one needs to know the $s$-dependent ensemble-averaged total cross section $\Sigma_t(s)$, or its corresponding path-length distribution function $p(s)$. We consider a 1-D spatially periodic system consisting of alternating solid and void layers, randomly placed in the infinite line. In this preliminary work, we assume transport in rod geometry: particles can move only in the directions $\mu=\pm 1$. We obtain an analytical expression for $p(s)$, and use this result to compute the corresponding $\Sigma_t(s)$. Then, we proceed to solve the non-classical equation for different test problems. To assess the accuracy of these solutions, we produce "benchmark" results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the results obtained with the non-classical equation accurately model the ensemble-averaged scalar flux in this 1-D random system, generally outperforming the widely-used atomic mix model. We conclude by discussing plans to extend the present work to slab geometry, as well as to more general random mixtures.