Back in 1855 a mathematical physicist turned doctor, called Adolf Eugen Fick, created what is now known as Fick’s laws of diffusion. The first is simply that the the diffusive flux is proportional to the concentration gradient and the second is that the rate of concentration change is proportional to the derivative of the concentration gradient. Later the same conclusion was reached when applying the method of random walks to particles in suspension and the final result was reached where the mean squared displacement of a particle is proportional to the time since it stared to move. This is considered the standard description of the diffusion process:

**〈 χ**^{2 }〉 ∝ t

However there are some fluids that refuse to follow this pattern especially if they would be considered turbulent. In these cases it is often more accurate to write:

**〈 χ**^{2 }〉 ∝ t^{α}

Where the alpha takes values in the range of zero to two. When it is greater than one the process is described as superdiffusion and when lower than one it is named subdiffusion. Examples of these anomalous diffusions have been found in viscoelastic substances, chaotic plasmas and transport through porous media.

It is also possible to apply such diffusion laws to that of light. When photons travel through a media and don’t get absorbed they can instead be repeatedly scattered. Since the distance between each scattering is a random walk, when there are enough photons they behave in accordance with the diffusion equations. This paper has taken to studying pulses of light in one dimension of dielectric materials and has successfully presented the ability to change the diffusion type from sub to superdiffusion. It was also shown that the the displacement law for photons in the media was:

**〈 χ**^{2 }〉 ∼ log^{β }(t)

Diffusion of this form is called ultra slow Sinai diffusion (after the person who suggested it) and has only been previously described by theory making this the first observed example of the effect. Proving its existence means that it could now be used to produce strange propagation and wave dispersion in optical devices.

Paper links: Fractional Transport of Photons in Deterministic Aperiodic Structures

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