RADIATION TRANSPORT THROUGH RANDOM MEDIA REPRESENTED AS MEASURABLE FUNCTIONS: POSITIVE VERSUS NEGATIVE SPATIAL CORRELATIONS

摘要

We investigate particle transport through media with randomly variable material properties with a special emphasis on very high spatial frequencies. In multi-dimensional transport theory, particularly in computational studies, collision coefficients are assumed to be "functions" in the conventional sense of the word: they have specific values at every point in the medium. This assumption actually constrains spatial variability in the high-frequency range in a manner that is not a mathematical necessity and relaxing it opens the door to interesting transport physics. From the standpoint of the integral transport equation, coefficient variations in space can indeed be represented as "measures," I.e., mathematical constructs that only make numerical sense under an integral (in the sense of Lebesgue). The new results are based on a variability model consisting of a zero-mean Gaussian scaling noise riding on a constant value large enough with respect to the amplitude of the noise to yield overwhelmingly non-negative coefficients; technically, this represents a "measurable" function. We first generalize known results about sub-exponential transmission from regular (almost everywhere continuous) to merely measurable functions with positively-correlated fluctuations. Another interesting outcome is that, in this very general measure-based framework, one can use conventional (continuum-limit) transport theory to address negatively- as well as positively-correlated stochastic media. We thus resolve a controversy concerning recent claims that only discrete-point process approaches can accommodate negative correlations, I.e., anti-clustering of the obstructing material particles. We obtain in this case the predicted super-exponential behavior, but only over a limited range of transport distances.