Asymptotic theory of the linear transport equation in anisotropic media

Abstract

We consider linear transport in an anisotropic medium with velocity dependent cross sections σ(r,v,t) and scattering kernel P(r,v′→v,t). We introduce a scaling in terms of a small parameter ϵ, where the leading-order term describes an equilibrium in velocity space between collisions with a cross section that is an even function of v and scattering modes even-even and odd-odd in v and v′. We show that the asymptotic solution of the transport equation leads to a diffusion equation with a drift term with an error in ϵ2 and derive consistent initial and boundary conditions from the analysis of the initial and boundary layers. The analysis of the drift terms shows that they result from anisotropic interactions with the medium and also from streaming between neighboring but different equilibria. The restriction of our results to isotropic media yields back the Larsen–Keller diffusion equation, while the one-speed form reduces to the result obtained by Pomraning and Prinja [Ann. Nucl. Energy 22, 159 (1995)] for the particular case of isotropic cross sections with an “output” scattering kernel P(r,Ω,t).