A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

TL;DR

This paper extends classical radiative transfer theory by developing unbiased Monte Carlo estimators and diffusion approximations for generalized scenarios where scattering centers are spatially dependent, providing benchmarks and new analytical tools.

Contribution

It introduces generalized collision and track-length estimators for GRT and derives Green's functions and diffusion approximations for isotropic point sources in infinite media.

Findings

Derived four Green's functions for isotropic point sources in GRT.

Developed moment-preserving diffusion approximations involving first four moments.

Provided benchmark solutions for Monte Carlo estimators in GRT.

Abstract

When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to angular flux / radiance because the macroscopic cross-section $\Sigma_t(s)$ depends on the distance $s$ to the previous collision or boundary. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.