# Quantification of divergence in real and apparent Monte Carlo variances near reflective boundaries

## Abstract

Monte Carlo methods, used to model nuclear systems, have been observed to underpredict the statistical variances in calculated results, a phenomenon which is magnified when tallied near reflecting boundaries. This underprediction is due to the fact that the histories used to calculate Monte Carlo results are often correlated to each other, negating the underlying assumption in the apparent variance calculation and introducing a bias. The real variance can be calculated using various statistical methods, and the difference between real and apparent variance quantifies the magnitude of the underprediction of variance. Although methods exist to calculate the real variance in Monte Carlo results, the physical drivers of the under-prediction phenomenon are still poorly understood. This work presents a study of the behavior of Monte Carlo variances near reflecting boundaries as a function of the fundamental physical parameters of the system. Using one-speed nuclear data with a one-dimensional slab model, the ratio of real to apparent variance is calculated for a wide range of slab thicknesses and cross sections. A nonlinear regression is then applied to these ratios to generate a fitting function that quantifies their relationship to the fundamental parameters. Analyzing this fitting function suggests that the variance underestimation can be expressed as a function of the slab thickness in mean free paths and the absorption probability per interaction. These results indicate that the behavior of variances near reflecting boundaries may be dependent on the number of boundary crossings a particle undergoes before interacting.