A Kinetic Theory for Nonanalog Monte Carlo Particle Transport Algorithms: Exponential Transform with Angular Biasing in Planar-Geometry Anisotropically Scattering Media

Abstract

We show that Monte Carlo simulations of neutral particle transport in planar-geometry anisotropically scattering media, using the exponential transform with angular biasing as a variance reduction device, are governed by a new “Boltzmann Monte Carlo” (BMC) equation, which includes particle weight as an extra independent variable. The weight moments of the solution of the BMC equation determine the moments of the score and the mean number of collisions per history in the nonanalog Monte Carlo simulations. Therefore, the solution of the BMC equation predicts the variance of the score and the figure of merit in the simulation. Also, by (i) using an angular biasing function that is closely related to the “asymptotic” solution of the linear Boltzmann equation and (ii) requiring isotropic weight changes at collisions, we derive a new angular biasing scheme. Using the BMC equation, we propose a universal “safe” upper limit of the transform parameter, valid for any type of exponential transform. In numerical calculations, we demonstrate that the behavior of the Monte Carlo simulations and the performance predicted by deterministically solving the BMC equation agree well, and that the new angular biasing scheme is always advantageous.