Proof that Combining the Forced-collision and DXTRAN Monte Carlo Variance-reduction Techniques is Fair
This report provides mathematical proof that the MCNP DXTRAN (also known in other codes as forced-flight) and forced-collision variance-reduction techniques do not bias the expected value of Monte Carlo simulation estimates when combined. Proof is also provided that the techniques are unbiased when used independently. To prove that the techniques are unbiased, this report derives the first History Score Moment Equation (HSME) for non-multiplying media employing only forced collisions, only DXTRAN, and then both the variance-reduction techniques combined, as defined next. The HSMEs are found by forming the History Score Probability Density Functions (HSPDFs) and then taking the first score moment. Following its derivation, each HSME with a variance-reduction technique employed is reduced to the HSME for an analog simulation. Because the HSME represents the expected contribution to estimators in the simulation, reducing the HSME with variance reduction to the analog HSME shows that the simulation is unbiased despite the variance-reduction technique considered in the HSME. Analysis regarding higher-score moments of each technique is the subject of prior work and is not addressed herein. Throughout this work, the phase space p is defined to be the particle position x, direction-of-flight unit vector $\hat{Ω}$, energy E, and statistical weight w, $$p ≡ (x; \hat{Ω}; E; w).$$ A reduced phase-space excluding the statistical weight of the particle, $$r ≡ (x; \hat{Ω}; E),$$ is also used.