Accurate reaction-diffusion limit to the spherical-symmetric Boltzmann equation

We resolve a long-standing question regarding the suitable effective diffusion coefficient of the spherical-symmetric transport equation, which is valid at long times. To that end, we generalize a transport solution in three dimensions for homogeneous media, to include general-collisional properties, including birth-death events and linearly anisotropic scattering. This is done by introducing an exact scaling law relating the Green's function of the pure-scattering case with the general-collision case, which is verified using deterministic and Monte Carlo simulations. Importantly, the effective diffusion coefficient is identified by inspecting the transport solution at long times.