Diffusion in pore fractals: A review of linear response models

A major aspect of describing transport in heterogeneous media has been that of relating effective diffusivities to the topological properties of the medium. While such effective transport coefficients may be useful for mass fractals or under steady state conditions, they are not adequate under transient conditions for self-similar pore fractal media. In porous formations without scale, diffusion is anomalous with the mean-squared displacement of a particle proportional to time raised to a fractional exponent less than unity. The objective of this review is to investigate the nature of the laws of diffusion in fractal media using the framework of linear response theory of nonequilibrium statistical mechanics. A Langevin/Fokker-Planck approach reveals that the particle diffusivity depends on its age defined as the time spent by the particle since its entry into the medium. An analysis via generalized hydrodynamics describes fractal diffusion with a frequency and wave number dependent diffusivity.