Abstract
As an illustration of anomalous diffusion in sparsely connected systems we study the properties of a model of porous media characterized by a random fracture/pore network. The structural connectivity of the system is furnished by a long-ranged bond percolation model and transport throughout it is described by a corresponding long-tailed distribution of crossing times. Both spatial and temporal problems are expressed in the language of random walks, and we introduce a renormalization group (RG) transformation for the distributions and obtain as a fixed point the Weierstrass walk. This walk exhibits a transition between Gaussian to fractal behavior in either spatial or temporal representations, and we analyze the conditions for the fractal regime that results in dispersive diffusion. Ordinary and generalized entropy functions decrease along the RG transformation flows and attain minima at the fixed points. We use the Tsallis entropy index q to measure non-extensivity as behavior departs from Gaussian.
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