A Reciprocal Formulation of Nonexponential Radiative Transfer. 1: Sketch and Motivation

Abstract

Previous proposals to permit nonexponential free-path statistics in radiative transfer have not included support for volume and boundary sources that are spatially uncorrelated from the scattering events in the medium. Birth-collision free paths are treated identically to collision–collision free paths and application of this to general, bounded scenes with inclusions leads to nonreciprocal transport. Beginning with reciprocity as a desired property, we propose a new way to integrate nonexponential transport theory into general scenes. We distinguish between the free-path-length statistics between correlated medium particles and the free-path-length statistics beginning at locations not correlated to medium particles, such as boundary surfaces, inclusions, and uncorrelated sources. Reciprocity requires that the uncorrelated free-path distributions are simply the normalized transmittance of the correlated free-path distributions. The combination leads to an equilibrium imbedding of a previously derived generalized transport equation into bounded domains. We compare predictions of this approach to Monte Carlo simulation of multiple scattering from negatively correlated suspensions of monodispersive hard spheres in bounded two-dimensional domains and demonstrate improved performance relative to previous work. We also derive new, exact, reciprocal, single-scattering solutions for plane-parallel half-spaces over a variety of nonexponential media types.

Keywords:

• Linear transport theory
• kinetic theory
• generalized linear Boltzmann equation
• reciprocity
• stochastic media

Acknowledgments

We would like to thank Johannes Hanika and the anonymous reviewers for their recommendations for improving the article.

Table 1. Summary of our notation and relationships between quantities in our formalism.