Application of a Discretized Phase-Space Approach to the Analysis of Monte Carlo Uncertainties

In the study of Monte Carlo statistical uncertainties for iterated-fission-source calculations, an important distinction is made between the real and the apparent variances. The former is the actual variance of a Monte Carlo calculation result, while the latter is an estimate of the former obtained using the results of the fission generations in the formula for uncorrelated random variates. For years it has been known that the apparent variance is a biased estimate of the real variance, and the reason for the bias has been understood. More recently, several authors have noted various interesting phenomena regarding the apparent and the real variances and the relationships among them. Some of these are an increase in the apparent variance near surfaces with reflecting boundary conditions, a nonuniform spatial distribution of the ratio of the apparent-to-real variance, the dependence of this ratio on the size of the region over which the result is tallied, and a rate of convergence of the real variance that is less than the inverse of the number of neutron histories run. This paper discusses a theoretical description of the Monte Carlo process using a discretized phase-space and then uses it to explain the causes of these phenomena.