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. 
Richard Vasques and Kai Krycki
cond-mat.dis-nn math-ph math.MP nucl-th