Asymptotic constraints for 1D planar grey photon diffusion from linear transport with special-relativistic effects

We derive a grey linear diffusion equation for photons with respect to inertial (or lab-frame) space and time, using asymptotic analysis in 1D planar geometry. The solution of the equation is the comoving radiation energy density. Our analysis does not make use of assumptions about the magnitude of velocity; instead we derive an asymptotic scaling in the lab frame such that we avoid apparent non-physical pathologies that are encountered with the standard static-matter scaling. We permit the photon direction to be continuous (as opposed to constraining the analysis to discrete ordinates). The result is a drift-diffusion equation in the lab frame for comoving radiation energy density, with an adiabatic term that matches the standard semi-relativistic diffusion equation. Following a recent study for discrete directions, this equation reduces to a pure advection equation as the velocity approaches the speed of light. We perform preliminary numerical experiments comparing solutions to relativistic lab-frame Monte Carlo transport and to the well-known semi-relativistic diffusion equation.

💡 Research Summary

The paper presents a rigorous derivation of a grey (frequency‑averaged) photon diffusion equation that remains valid in the presence of special‑relativistic bulk motion, using asymptotic analysis in a one‑dimensional planar geometry. Starting from the laboratory‑frame linear transport equation with isotropic scattering, the authors first transform the scattering kernel into the lab frame by assuming isotropic, elastic scattering in the comoving frame and by explicitly accounting for Doppler frequency shifts. This yields a kernel that depends on the Lorentz factor γ and the velocity parameter β=v/c through factors of the form γ(1−βμ)³(1−βμ′)², where μ and μ′ are the incoming and outgoing direction cosines in the lab frame.

To handle the algebraic complexity introduced by repeated Lorentz transformations, the authors introduce a family of functions λₙ,ₖ(μ)=μᵏγⁿ(1−βμ)ⁿ and their integrals Λₙ,ₖ. These functions possess simple product, differentiation, and recursion properties that streamline the asymptotic expansion. The intensity ψ and its comoving counterpart ψ₀ are expanded in powers of a small parameter ε, while the Lorentz transformation relation ψ=λ₄,₀(μ)ψ₀ is imposed at each order.

Two scaling strategies are examined. The first adopts the standard comoving‑frame diffusion scaling (B(μ)=0) and quickly reveals that the leading‑order scalar equation does not close; non‑physical terms remain at O(ε²). The second introduces an anisotropic scalar multiplier B(μ)=βμ, which effectively captures the coupling between spatial gradients and bulk motion. With this modified scaling, the O(ε²) equations close, yielding a drift‑diffusion equation for the comoving radiation energy density E₀ expressed entirely in laboratory‑frame coordinates:

∂E₀/∂t + ∂(βcE₀)/∂x = ∂/∂x