When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. 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 and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
In classical linear transport, various equivalent mathematical approaches are loosely divided into two camps @cite_146 @cite_93 based on whether or not the focus is on the history of a single particle (random flight) or on a single volume element (Boltzmann). The nonexponential random flights of GRT have been studied since the early 1900s @cite_132 @cite_8 . The related generalized linear Boltzmann equation (GLBE) for GRT was more recently proposed and the connection to the related integral equation of random flight established @cite_133 @cite_39 . Existence and uniqueness of solutions for the GLBE have been established @cite_104 and the GLBE has been extended to support anisotropic random correlated media @cite_62 with direction-dependent cross sections. Other Boltzmann-like equations have appeared @cite_49 @cite_33 @cite_78 @cite_35 , and some equivalences between them have been established @cite_87 . The most relevant aspect of these works to the present is not the integro-differential form of the transport equation, however, but the formalism of total macroscopic cross sections @math with a free-path-length memory variable @math . Through this lens, nonexponential random flights take on a familiar linear-transport character, and the subtleties of collision rate and flux balance and generalization of familiar Monte Carlo methods become more evident.
@cite_35 @cite_62 @cite_33 @cite_78 @cite_8 @cite_132 @cite_87 @cite_39 @cite_104 @cite_146 @cite_133 @cite_49 @cite_93
2081489139 2107648085 2123942277 1844244698
Abstract This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications. We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation for the stationary collision density and scalar flux about an isotropic point source in an infinite d-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially distributed free paths in arbitrary dimensions as well as a number of nonclassical transport theories (nonexponential random flights) that describe a broader class of transport processes within partially correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean’s moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean’s approach produces compact, analytic approximations that are, overall, more accurate for high absorption and for smal... In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt’s model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE. Abstract This paper extends a recently introduced theory describing particle transport for random statistically homogeneous systems in which the distribution function p ( s ) for chord lengths between scattering centers is non-exponential. Here, we relax the previous assumption that p ( s ) does not depend on the direction of flight Ω ; this leads to a new generalized linear Boltzmann equation that includes angular-dependent cross sections, and to a new generalized diffusion equation that accounts for anisotropic behavior resulting from the statistics of the system.