Collision expansion for the radiative transport equation: Analytical results and numerical simulations

We consider the collision expansion of the Green's function of the radiative transport equation (RTE) in an infinite medium. Analytical expressions in terms of quadratures of the most simple form are given for all orders of the expansion. Singularities of the Green's function are considered in detail. While it is well known that the zeroth and first terms in the expansion are singular (and proportional to delta functions), we show that the second order term contains a logarithmic singularity. All higher-order terms are regular. We further establish a relation between the Green's function and the signal measured by a collimated detector. In the presence of singularities, this relation is not always obvious and, at second order, it cannot be stated in a form that is independent of the acceptance angle of the detector. We also consider the density and energy current. Theoretical results are supported by Monte-Carlo simulations.