On calculating self-collision probabilities

A numerical method for calculating first-flight collision, escape and transmission probabilities in three dimensions has been proposed by the author. The method consists of a subdivision of the spatial zones that constitute the domain under study, and where the flat-flux approximation is used, into parts called elements and an assumption that the contribution from a source element to a sink element can take place only along the path that joins their centers of mass. The resulting expressions are evaluated with the number of elements per zone increased successively and Richardson extrapolation to infinite number of elements applied to the sequence of results, until convergence to within a desired degree of accuracy in the probabilities is attained. In this work, a critical aspect of the method, namely the fact that the contribution from an element to itself is not taken into account in the self-collision probability calculation, is examined in detail. A mathematical justification for this approximation is offered and the ensuing numerical error for the case of a cube is evaluated with the introduction of a third level of discretization in the calculation.