Change of Scale in Physics of Random Media

Problems of change of scale in physics of random media handled in this chapter concern the prediction of the macroscopic behavior of a physical system from its microscopic behavior. Most physical properties of random media are concerned by this approach, like for instance the dielectric permittivity ϵ in electrostatics, the permeability K or flows in porous media, or elastic moduli of composites. This topic concerns the definition an calculation of effective (or macroscopic) properties of an equivalent homogeneous medium from lower scale information by homogenization tools. In a first step, models of RS or of RF are used to represent maps of properties on a microscopic level. Then, the homogenization problem is addressed by a perturbation expansion of the physical fields of interest and by the calculation of bounds of effective properties for models of random structures by means of correlation functions of local properties. This illustrated in the case of linear constitutive behavior by third order bounds and their application to models of RF and of random sets studied in previous chapters. Finally local fluctuations of fields on a point level are characterized by their second order moments derived from the effective properties.