Fractional radiative transport in the diffusion approximation

Abstract

The fractional radiative transport equation describes the propagation of particles, whose path length distribution $$p(\ell )=-\partial _\ell E_\alpha (-\sigma _t\ell ^\alpha )$$ satisfies the generalized Lambert–Beer law $$\begin{aligned} \partial _\ell ^\alpha p(\ell )=-\sigma _t p(\ell ),\quad \ell >0, \end{aligned}$$ for $$\alpha \in (0,1]$$ with $$E_\alpha (x)$$ being the Mittag-Leffler function and $$\sigma _t$$ is the total attenuation coefficient. Within the classical radiative transport theory the diffusion equation is known to be as the most often considered approximation. In this paper, we derive a generalized diffusion model on the basis of the fractional radiative transport equation using the Fourier transform in combination with the Pade approximation method. Moreover, the associated diffusive flux vector is given in form of a generalized Fick’s law containing the fractional gradient operator. It is shown via comparisons to the Monte Carlo method and an exact analytical solution that the derived diffusion approximations agree quite well with the exact solution of the fractional radiative transport equation even for highly absorbing media.