Dynamic equations for propagation of light through dust For smoothly inhomogeneous media light propagation can be modeled by Maxwell's equations with non-constant $\varepsilon$ and $\mu$. This allows to relatively easily model propagation in the medium with varying index of refraction. But suppose a medium with refraction index $n_0$ is randomly seeded with huge number of grains with $n_1$, with some density distribution $\rho(\vec r)$, and the light wavelength is much smaller than the grain size. This would correspond to propagation of light through inhomogeneous dust or fog.
My question: is there any equation, which under these conditions models the dust as a solid medium and allows to compute propagation of light in such a medium? It seems there should be something based on geometric optics, but working in terms of light intensity instead of individual rays. Maybe some sort of diffusion equation?
 A: Much of scattering theory falls under the heading of Mie scattering. It examines how light scatters off uniform spheres with a given electromagnetic permittivity. In fact, it provides exact solutions to Maxwell's equations in this case. The original work was done in Mie 1908 (in German). Various further approximations can be made, leading to things like Rayleigh scattering in the long-wavelength limit.
Short wavelengths would indeed be more akin to ray tracing. Note, though, that you would have to have some rather large particles to be considered much larger than the wavelength of optical light. Also, Mie theory (which always works, but which will require more and more terms in its series solution for smaller and smaller wavelengths) provides an interesting result here: the extinction (scattering plus absorption) cross section asymptotically approaches twice the geometric cross section expected from ray tracing, due to diffraction effects. This is the "extinction paradox", though there is nothing particularly mysterious about it.
The general idea is Mie theory (or something more advanced, like the discrete dipole approximation) gives you numbers for absorption -- characterized by an optical depth $\tau_1$ -- and how much light is scattered out of its original trajectory -- characterized by the "degree of elongation of the scattering indicatrix" $\tilde{\omega}_1$. Then you can apply this to get a flux diffusion equation, as is done by Piotrowski 1956 and 1961. Piotrowski was particularly interested in sunlight scattering through clouds, and he notes that the equation of interest depends only on $\tau_1 (1 - \tilde{\omega}_1/3)$.
In the limit that each particle interaction scatters the photon uniformly across the whole sphere, $\tilde{\omega}_1 \to 0$ and diffusion only depends on $\tau_1$. This is to be expected, since it is the same as the photon being absorbed and re-emitted with each interaction.
More gory details can be found in Irvine 1965. Roughly:


*

*One's microscopic scattering theory returns some function $\Phi$ (essentially $\tilde{\omega}_1$ from before) of cosines of scattering angles.

*$\Phi$ is integrated over an angle to yield $F$.

*The source function ("external source function" in that paper) $J_1$ is defined from $F$ and $\tau$.

*Two coupled equations are solved, relating specific intensity $I$, its derivative with respect to $\tau$, $J_1$, and $J$ (a convolution of $F$ and $I$).

