Lévy flights

The standard diffusion approximation has its roots in the celebrated Central Limit Theorem. It doesn't really matter the microscopic nature of scattering or the form of the cross section, macroscopically  the  step lenght distribution will always converge to a gaussian. The central limit theorem fails to be valid when the  lenght distribution of a single step does not have a finite variance. In this case the diffusion equation cease to be valid and the system becomes superdiffusive.
Not much is known yet about the properties of superdiffusive transport and the fractional diffusion equation (that describes this regime) do not even allow for a simple inclusion of boundary conditions.

Lévy glass
In order to study the properties of superdiffusion of light we developed a new class of material that we dubbed Lévy glasses. In these systems the density of scatterers is strongly inhomogeneous due to  the inclusion of a well-chosen distribution of index-matched spheres in the scattering medium.