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

TL;DR

Relativistic diffusion of photons in optically thick, moving media challenges classical diffusion due to superluminal signal propagation. The authors derive a fully relativistic 1D planar grey diffusion equation in the lab frame by applying targeted asymptotic scalings to the transport equation, introducing a velocity-derivative correction that yields a closed drift-diffusion form and preserves parity as $\\beta\to1$. The resulting diffusion operator scales as $D \sim 1/\\gamma^3$ and reduces to pure advection in the high-velocity limit, with improved agreement to lab-frame Monte Carlo transport over the semi-relativistic diffusion in high-$\\beta$ regimes; pathologies of simpler scalings are resolved by an anisotropic scaling and a non-acceleration condition. The framework enables stable DDMC/hybrid implementations and paves the way for extensions to 3D and frequency-dependent radiative transfer in relativistic flows, providing a practical bridge between transport theory and diffusion in relativistic astrophysical contexts.

Abstract

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.

Figures (5)

• Figure 1: Polar graphs of stability measure $\mathcal{G}(\theta,\alpha)$ for $\beta=0.6$ versus angle $\theta=K\Delta x$ (green dotted). Time centering is $\alpha=0$ (explicit Euler; left panel) or $\alpha=0.5$ (Crank-Nicolson; right panel). For comparison, the unit circle (blue dashed) and $\mathcal{G}(\theta,\alpha)$ with $\beta=0$ (orange solid) are also plotted.
• Figure 2: Scaled scalar flux versus spatial coordinate for several values of constant $\beta$. The maximum value of MC transport (solid) is set to 1, and semi-relativistic (dashed) and fully relativistic diffusion (dotted) are scaled accordingly.
• Figure 3: Scaled scalar flux versus spatial coordinate for several values of $\Delta\beta=0.06,0.6$ in $\beta = 0.6 + (x-1/2)\Delta\beta$. The maximum value of MC transport (solid) is set to 1, and semi-relativistic (dashed) and fully relativistic diffusion (dotted) are scaled accordingly.
• Figure 4: Scaled scalar flux versus spatial coordinate for several values of $\Delta\beta=0.06,0.6$ in $\beta = 0.3 + \Theta(x-1/2)\Delta\beta$. The maximum value of MC transport (solid) is set to 1, and semi-relativistic (dashed) and fully relativistic diffusion (dotted) are scaled accordingly.
• Figure 5: Example diagram of triangular recursion given by Eq. \ref{['eq7:app1']}. The first layer of orange rectangles is evaluated with Eqs. \ref{['eq2:app1']} and \ref{['eq4:app1']}. This particular diagram generates the factors needed for evaluating lab-frame $\mu$-weighted integrals from the corresponding comoving $\mu_0$-weighted integrals, up to linear anisotropy in the comoving frame. The red dashed line separates the factors needed for isotropic comoving intensity (left) from those needed for linear anisotropic comoving intensity (right).