It is the first review on the emerging topic of fractional dynamical equations. It presents the mathematical foundation as well as a discussion of the underlying physical ideas.

  Does it describe a new discovery or a new methodology that's useful to others?

Equations such as the fractional Fokker-Planck equation are a direct generalization of the standard equations for stochastic processes, the latter being the basis for modelling in a very broad range of systems in physics, chemistry, and many other areas. As power-laws are a fairly ubiquitous signature of many complex systems, the fractional generalization is expected to find similarly broad application. As such, fractional transport equations are a new methodology indeed, offering a tool to include external potentials to anomalous transport features, and their solution.

What happens when a little molecular cargo is transported in a biological cell, or a chemical is spilled and gets into the aquifer? Under certain conditions, they will not spread over space in the course of time, as we are used by the motion of a car or the spreading of a drop of cream in a cup of tea. They are going to behave anomalously, in most cases slower than one might naively expect. In fact so slow, that, for instance, one cannot define a characteristic breakthrough time for the spilled chemical to pass a certain point a distance away from the spillage. Processes like these can often be well described with the framework of fractional equations.

During my postdoc time, we wanted to find out how existing frameworks for describing anomalous diffusion could be extended to cases where external potentials cannot be neglected. We realized that the fractional Fokker-Planck equation is an outstanding tool for obtaining information about such systems.