Anomalous diffusion: Direct simulations and fractional calculus models

The classic hall-mark of a diffusion transport process is that the  length-scale of the spreading of a conserved quantity (heat, solute,  etc) changes with the square root of time. This phenomena is readily  observed in homogeneous materials. When appropriate heterogeneity is  present, however, the classic signal can be replaced with an anomalous  (non-local) behavior where power-law changes of the diffusion  length-scale with time differs from the square root. There are two, parts to that talk. In the first (based on collaborative work with Diogo Bolster and Fabio Reis) I will use direct simulations of 2-D  diffusion processes to demonstrate how the presence of heterogeneity can induce anomalous signals. The second part of the talk will focus  on fractional calculus models of anomalous diffusion. We will show, through scaling, how a one-dimensional diffusion model, with  fractional transient and flux terms produces anomalous transport  signals. Drawing from on going work with Namba and Rybka, I will also  comment on the physical validity of the fractional flux, with  particular focus on boundary conditions in finite domains.