Renormalization-group approach to noncoherent radiative transfer

In noncoherent transfer, a photon undergoes multiple scattering with frequency changes, the frequency profile $\phi \left(0\mathrm{}\le \phi \le \Lambda \right)$ after each scattering is prescribed, the mean free path varies like ${\phi }^{-1\mathrm{}}$, and there is a small probability of destruction $\epsilon$. In contrast with the coherent (monochromatic) case, diffusive behavior is not obtained as $\epsilon \to 0$. The scaling law of the intensity of radiation can be obtained with a technique suggested by the renormalization-group approach to critical phenomena. The equation of transfer for "core" photons (say, $\Lambda \ge \phi >\frac{1}{2\Lambda }$) is solved in terms of "wing" photons ($\frac{\Lambda }{2}\ge \phi >0\mathrm{}$). The solution is substituted wherever core photons appear in the equation for the wing photons. A closed equation for wing photons is thus obtained. After rescaling of the variables (wave number and $\phi$) it resembles the original equation ($\phi$ again runs from 0 to $\Lambda$) but has a nonlocal scattering operator. The procedure is iterated and the successive equations are found to approach a fixed form with suitable choice of the rescaling factors. This choice fixes the scaling law. It depends only on the asymptotic behavior of the profile at large frequencies. Known results for the dependence of the "thermalization length" on $\epsilon$ are recovered.