Transport phenomena in fractal and heterogeneous media—input/output renormalization and exact results

The renormalization of transport equations and the exact solution of linear and nonlinear transport equations on fractals are discussed. Discrete balance equations are renormalized on a family of graphs parametrized with respect to the order of iteration. The analogies and differences between input/output renormalization of the Green functions in the case of transport phenomena and in the solution of wave propagation (Schrödinger equation, transmission lines) are analyzed. Exact results for sorption on fractals and for first-order reaction/diffusion kinetics are presented. The extension of this method to solve complex transport schemes (linear and nonlinear) is also discussed.