# Radiative transfer in half spaces of arbitrary dimension

## TL;DR

This paper generalizes classical radiative transfer problems to arbitrary Euclidean dimensions, deriving solutions and properties that reveal universal behaviors and dimension-dependent differences in scattering and diffusion modes.

## Contribution

It introduces a unified framework for radiative transfer in any dimension, extending classical 3D solutions to higher and lower dimensions with closed-form expressions.

## Key findings

- Universal properties invariant to dimension d
- Discrete diffusion mode not universal for d > 3 in absorbing media
- Unexpected correspondences between different dimensions and scattering types

## Abstract

We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain $\mathbb{R}^d$ for arbitrary $d \ge 1$. A continuous family of pseudo-problems and related $H$ 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 $d$ are highlighted and we find that a discrete diffusion mode is not universal for $d > 3$ in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.