Anomalous dispersion, renormalization groups, scaling laws and classification: A reflection on recent efforts

Motivated by the need to understand the dynamics of motile particles in porous media, our team has applied renormalization group techniques to both upscale and classify anomalous dispersive/diffusive behavior. Central limit theorems, which lead to a specific type of renormalization group, are employed in several cases to upscale transport of motile particles in porous media that display a specific type of fractal character. The old standby classification for diffusion (which we use interchangeably with dispersion) says a particle is anomalous if its mean square displacement is not linear in time. This physically intuitive concept, is shown to be inadequate, and so is replaced by a scheme that relies on the fixed points of specific renormalization group operators. Various asymptotic limits are examined, and scaling laws for the limits are derived. A random renormalization operator is introduced for processes with multiple asymptotes and unknown self-similarity index, and a Bayesian tool is employed to obtain scaling laws that are weighted averages of power laws.