Isotropic Scattering in a Flatland Half-Space

We solve the Milne, constant-source and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener-Hopf method. The Flatland H-function is derived and benchmark values and some identities unique to Flatland are presented. A number of the derivations are supported by Monte Carlo simulation. 1. Intro The study of linear transport theory [1, 2] in lower-dimensional spaces serves a number of purposes. The simplicity of the one-dimensional rod model [3] makes it a useful tool for education [4] and occasionally the starting place for exploring new general transport processes [5]. Most rod model problems can be solved exactly and admit simple closed-form solutions, with diffusion encompassing the entire solution. These properties are attractive, but distance the rod model from the complexity of three-dimemsional transport and therefore also limit its utility. Sandwiched between the rod model and traditional three-dimemsional scattering, two-dimensional “Flatland” provides a transport domain with much of the complexity of full 3D scattering, while ocassionally admitting simple closedform solutions that have not been found in 3D (interestingly, the time-resolved Green’s functions for the isotropic point source in infinite media are known exactly for 2D and 4D, but not 3D [6]). Because Flatland transport research in bounded media has led to insights that improve the efficiency of 3D light transport (albeit so far participating media has not been considered [7]), we solve the classic half space problems in Flatland for isotropic scattering and investigate the form of the Flatland H function and some of its numerical properties. These derivations may aid future studies of this form in many fields. These solutions may also directly apply to physical processes where the transport is fundamentally two-dimensional [8, 9, 10, 11]. 1.1. Related Work Infinite media problems have been well studied in Flatland as well as spaces of general dimension [12, 13, 14, 15, 16, 6, 17, 18, 19] and for beams [20]. Some Preprint submitted to Elsevier February 7, 2018 ar X iv :1 80 2. 02 12 0v 1 [ ph ys ic s. cl as sph ] 3 F eb 2 01 8 exact solutions have been presented for bounded [21] and layered [22] media, and the singular eigenfunctions for Flatland have been derived [23]. However, to the best of the authors’ knowledge, solutions to the classic Milne and albedo problems for the half-space and the H-function have not been presented. 2. General Theory The Flatland one-speed transport equation can be written [20] as cos θ ∂φ(x, y, θ) ∂x +sin θ ∂φ(x, y, θ) ∂y +φ(x, y, θ) = c 2π ∫ π −π dθ′φ(x, y, θ′)+ 1 2π S(x, y) (1) where c = Σs/Σ and the notation is standard. The two-dimensional analog of angular flux (or radiance) [7] is denoted φ. If we assume that there is spatial variation in only the x-direction we find cos θ ∂φ(x, θ) ∂x + φ(x, θ) = c 2π ∫ π −π dθ′φ(x, θ′) + 1 2π S(x). (2) We change the angular variable in Eq.(2) such that μ = cos θ, which leads to ( μ ∂ ∂x + 1 ) φ(x, μ) = c 2π 2 ∫ 1 −1 dμ′ √ 1− μ′2 φ(x, μ′) + S(x) 2π . (3) For the sake of completeness we now convert Eq.(3) to integral form for the scalar flux φ0(x) = 2 ∫ 1 −1 dμ′ √ 1− μ′2 φ(x, μ′). (4) Re-arranging Eq.(3) as ∂ ∂x ( φ(x, μ)e ) = 1 2πμ (cφ0(x) + S(x)) e x/μ (5) and for μ > 0 let us integrate from 0 to x, viz: φ(x, μ) = φ(0, μ)e−x/μ+ 1 2πμ ∫ x 0 dx′ (cφ0(x ′) + S(x′)) e−(x−x ′)/μ;μ > 0. (6) Assuming that we have a finite slab of width a we can now integate Eq.(5) from x to a for mu < 0, thus φ(x, μ) = φ(a, μ)e(a−x)/μ− 1 2πμ ∫ a x dx′ (cφ0(x ′) + S(x′)) e ′−x)/μ;μ < 0. (7)