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.
Going beyond 3D, higher-order dimensions occasionally find application in practice. Exact time-dependent solutions in 2D and 4D have been combined to approximate the unknown 3D solution for the isotropic point source in infinite media @cite_63 , and later applied to a time-dependent searchlight problem using the method of images @cite_23 . In the study of cosmic microwave background radiation it has even been considered to change dimension over the course of a single random flight @cite_83 .
@cite_23 @cite_63 @cite_83
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The Green’s function of the time dependent radiative transfer equation for the semi-infinite medium is derived for the first time by a heuristic approach based on the extrapolated boundary condition and on an almost exact solution for the infinite medium. Monte Carlo simulations performed both in the simple case of isotropic scattering and of an isotropic point-like source, and in the more realistic case of anisotropic scattering and pencil beam source, are used to validate the heuristic Green’s function. Except for the very early times, the proposed solution has an excellent accuracy (>98 for the isotropic case, and >97 for the anisotropic case) significantly better than the diffusion equation. The use of this solution could be extremely useful in the biomedical optics field where it can be directly employed in conditions where the use of the diffusion equation is limited, e.g. small volume samples, high absorption and or low scattering media, short source-receiver distances and early times. Also it represents a first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and diffuse optical imaging. Moreover the proposed solution can be useful to several research fields where the study of a transport process is fundamental. The time-dependent Boltzmann equation, which describes the propagation of radiation from a point source in a random medium, is solved exactly in Fourier space. An explicit expression in real space is given in two and four dimensions. In three dimensions an accurate interpolation formula is found. The average intensity at a large distance @math from the source has two peaks, a ballistic peak at time @math and a diffusion peak at @math (with @math the velocity and @math the diffusion coefficient). We find that forward scattering adds a tail to the ballistic peak in two and three dimensions, @math and @math , respectively. Expressions in the literature do not contain this tail. We shall study random flights that start in a space of one given dimension and, after performing a definite number of steps, continue to develop in a space of higher dimension. We show that if the difference of the dimension of spaces is even, then the probability density describing the composite flight can be expressed as marginalizations of the probability density associated to a random flight in the space of less dimensions. This dimensional reduction is a consequence of Gegenbauer addition theorem.