A Reciprocal Formulation of Non-Exponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation
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When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section Σ_t(s) depends on the distance to the previous collision s. We generalize collision, track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory. We consider collision estimators with additional ficticious scattering which, in the limit of infinite ficticious density, become track-length estimators. The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path distributions, which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner. To numerically evaluate the estimators we derive new Green's functions for the isotropic point source in infinite media with isotropic scattering, derived using a Fourier transform approach. Additionally, new moment-preserving diffusion approximations for non-exponential random media are derived for both correlated and uncorrelated emission, and their accuracy is evaluated for Gamma and power-law free-path distributions. The general imbalance of flux and collision rate is explored in a variety of scenarios.
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