Spectral Analysis of Stochastic Noise in Fission Source Distributions from Monte Carlo Eigenvalue Calculations

In this paper a new analysis of the characteristics and propagation of random fluctuations in Monte Carlo (MC) eigenvalue calculations is presented. In particular, random fluctuations in the fission source distribution produced from a fixed set of initial neutron source locations are considered. Unlike previous analyses, this work relies on spectral theory to analyze a general class of MC eigenvalue transport algorithms. This new analysis uses a simple delta function representation to express the set of discrete neutron source locations as a continuous function, which can be written in terms of the eigenfunctions of the transport operator. This allows the iterative MC source convergence process to be analyzed using existing techniques developed for the traditional power iteration method. Furthermore, it can be shown that the random fluctuations introduced during these source iterations are due to the fact that MC algorithms must approximate the continuous fission source distribution by a discrete set of randomly sampled fission sites. Based on this result, a series expansion for the MC fission source realization is derived, and the expected value of the expansion coefficients for this series are shown to converge to the expansion coefficients for the continuous fission source distribution. An expression for the variance of each of the MC fission distribution expansion coefficients is also presented. Numerical results for a simple 1-D slab geometry problem are presented to support the theoretical results derived in the paper.