MORPHOLOGY OF SPATIAL PATTERNS - POROUS-MEDIA, SPINODAL DECOMPOSITIONAND DISSIPATIVE STRUCTURES

The morphological characterization of patterns is becoming more and mo re important in Statistical Physics as complex spatial structures now emerge in many systems. A suitable family of morphological measures, k nown in integral geometry as Minkowski functionals, characterize not o nly the connectivity but also the content and shape of spatial figures . The Minkowski functionals are related to familiar geometric measures : covered volume. surface area? integral mean curvature, and Euler cha racteristic. Integral geometry provides powerful theorems and formulae which makes the calculus convenient for many models of stochastic geo metries, e.g. for the Boolean grain model. The measures are: in partic ular, applicable to random patterns which consist of non-regular, fluc tuating domains of homogeneous phases on a mesoscopic scale. Therefore , we illustrate the integral geometric approach by applying the morpho logical measures to such diverse topics as porous media, chemical-reac tion patterns, and spinodal decomposition kinetics: (A) The percolatio n threshold of porous media can be estimated accurately in terms of th e morphology of the distributed pores. (B) Turing patterns observed in chemical reaction-diffusion systems can be analyzed in tel ms of morp hological measures, which turn out to be cubic polynomials in the grey -level. We observe a symmetry-breaking of the polynomials when the typ e of pattern changes. Therefore, the morphological measures are useful order parameters to describe pattern transitions quantitatively. (C) The time evolution of the morphology of homogeneous phases during spin odal decomposition is described, focusing: on the scaling: behavior of the morphology, Integral geometry provides a means to define the char acteristic length scales and to define the cross over from the early s tage decomposition to the late stage domain growth.