Fluctuation Modeling of Macroscopic Cross Section with Spatial Correlation

Fluctuation modeling of the macroscopic cross section is studied in the framework of a continuously distributed stochastic medium. In particular, spatial correlation is approached by fractional Brownian motion (FBM) and randomized Weierstrass function (RWF). Here, FBM is capable of modeling correlation due to coordinate increments while RWF has the same property as FBM on a small scale, is able to confine the influence of correlation within a certain range of increments, and is globally under a fixed variance. In numerical experiments, first flights of neutral particles are examined using Woodcock tracking. Results obtained indicate that the attenuation of an uncollided beam becomes slower than the exponential law of the corresponding nonstochastic homogeneous medium as the spatial correlation changes from negative to positive; this departure to the slower side is very small or negligible in the full antipersistency limit of negative correlation. It is also shown that the departure from the exponential law of attenuation is nearly negligible if the influence of correlation is confined within the mean free path (mfp) determined by the macroscopic cross section of the corresponding nonstochastic homogeneous medium. However, the mfp's for individual realizations of the medium distribute widely. FBM turns out not to be feasible for modeling positive correlation. Overall, RWF virtually eliminates the risk of negative values of the macroscopic cross section inherent in the FBM modeling.