The types of anomalous diffusion given above allows one to measure the type, but how does anomalous diffusion arise? There are many possible ways to mathematically define a stochastic process which then has the right kind of power law. Some models are given here.
These are long range correlations between the signals continuous-time random walks (CTRW) and fractional Brownian motion (fBm), and diffusion in disordered media. Currently the most studied types of anomalous diffusion processes are those involving the following
* Generalizations of Brownian motion, such as the fractional Brownian motion and scaled Brownian motion
* Diffusion in fractals and percolation in porous media
* Continuous time random walks
These processes have growing interest in cell biophysics where the mechanism behind anomalous diffusion has direct physiological importance. Of particular interest, works by the groups of Eli Barkai, Maria Garcia Parajo, Joseph Klafter, Diego Krapf, and Ralf Metzler have shown that the motion of molecules in live cells often show a type of anomalous diffusion that breaks the ergodic hypothesis. This type of motion require novel formalisms for the underlying statistical physics because approaches using microcanonical ensemble and Wiener–Khinchin theorem break down.