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Rigorous asymptotic and moment-preserving diffusion approximations for generalized linear Boltzmann transport in arbitrary dimension

We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) 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 non-classical transport theories (non-exponential 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 small source-detector separations than either $P_1$ diffusion or rigorous asymptotic diffusion. These diffusion-based approximations are exact in the first two even spatial moments, which we derive explicitly for various non-classical transport types. We also discuss connections between the random-flight-theory derivation of the Green's function and the discrete spectrum of the transport operator and report some new observations regarding the discrete eigenvalues of the transport operator for general dimensions and free-path distributions.