text: AbstractWe 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 a...
  @cite_18: 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.
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  @cite_26: In this study, Green’s function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media.
  @cite_6: 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...
  @cite_3: The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.
  @cite_24: Abstract This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.
  @cite_19: We study transport and diffusion of classical waves in two-dimensional disordered systems and in particular surface waves on a flat surface with randomly fluctuating impedance. We derive from first principles a radiative transport equation for the angularly resolved energy density of the surface waves. This equation accounts for multiple scattering of surface waves as well as for their decay because of leakage into volume waves. We analyze the dependence of the scattering mean free path and of the decay rate on the power spectrum of fluctuations. We also consider the diffusion approximation of the surface radiative transport equation and calculate the angular distribution of the energy transmitted by a strip of random surface impedance.
  @cite_23: 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.
  @cite_5: In this paper we analyze some aspects of exponential flights , a stochastic process that governs the evolution of many random transport phenomena, such as neutron propagation, chemical biological species migration, or electron motion. We introduce a general framework for @math -dimensional setups, and emphasize that exponential flights represent a deceivingly simple system, where in most cases closed-form formulas can hardly be obtained. We derive a number of novel exact (where possible) or asymptotic results, among which the stationary probability density for 2d systems, a long-standing issue in Physics, and the mean residence time in a given volume. Bounded or unbounded, as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings.
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  @cite_13: Synopsis This paper deals with the solution of the generalized k -dimensional random flight problem, where the paths are distributed according to given probability functions and where the scattering collisions are non-isotropic. This is achieved by means of a new method which is largely based on recurrence relations and therefore very different from the well-known procedure using discontinuous integrals. In the special case, when the collisions are isotropic and the paths have given lengths, the new results become identical to those, which were originally obtained by the older method.
  @cite_25: In this study, the two-dimensional radiative transfer equation is solved considering the exact boundary conditions for describing the propagation of particles in bounded anisotropically scattering media. The obtained Green’s functions have an analytical dependence on the spatial and angular variables and, therefore, can be evaluated rapidly and implemented without any computational complexities. The Monte Carlo method was used for a successful verification of the derived solutions.
  @cite_20: 
summary: Infinite media problems have been well studied in Flatland as well as spaces of general dimension @cite_20 @cite_4 @cite_15 @cite_13 @cite_14 @cite_18 @cite_26 @cite_5 @cite_6 and for beams @cite_23 . Some exact solutions have been presented for bounded @cite_25 and layered @cite_24 media, and the singular eigenfunctions for Flatland have been derived @cite_3 . However, to the best of the authors' knowledge, solutions to the classic Milne and albedo problems for the half-space and the @math -function have not been presented. @cite_19 , using an asymptotic analysis, have presented solutions to the classic Milne and albedo problems in two dimensions. The solutions make use of the Flatland equivalent of Chandrasekhar's @math -function, which is left as the solution to an integral equation. We present a complementary derivation of the Milne and albedo problem solutions using the Wiener-Hopf technique. In addition, we present new solutions and benchmark values for @math and provide some Monte Carlo comparisons for the albedo problem.