We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain @math for arbitrary @math . A continuous family of pseudo-problems and related @math functions arises and includes the classical 3D solutions, as well as 2D "Flatland" and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension @math are highlighted and we find that a discrete diffusion mode is not universal for @math in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.
The study of transport problem in the case of general dimension can reveal how dimension @math impacts various aspects of the solution. New insights about 3D transport have been found by identifying correspondences between problems of differing configurations and dimensionalities @cite_52 @cite_31 @cite_30 , which often appear unexpectedly. Most investigations of this nature have considered only infinite media. In this work, we identify some new exact correspondences regarding anisotropic scattering in a 3D half space.
@cite_30 @cite_31 @cite_52
2227626537 2035369868 2528005577 2107648085
Abstract Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. We study the flatland (two dimensional) linear transport equation, under an angular2@pperiodicity assumption both on particle density function @j(x,y,@q) and on the differential scattering @s"s(@q). We consider the beam problem, with a forward peaked source on phase-space, and derive P"1 approximation with a diffusion coefficient of 1 2@s"t"r, (versus 1 3@s"t"r of the three dimensional problem), where @s"t"r is the transport cross section. Further assumptions as peaked @s"s(@q) near @q=0 (small angle of scattering), and small angle of flight (@q 0) yield Fokker-Planck and Fermi approximations with the diffusion coefficients @s"t"r (rather than @s"t"r 2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples. This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation. 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...