INTEGRAL GEOMETRY IN STATISTICAL PHYSICS

The aim of this paper is to point out the importance of a morphologica l characterization of patterns in Statistical Physics. Integral geomet ry furnishes a suitable family of morphological descriptors, known as Minkowski functionals. They characterize not only the connectivity (to pology) but also the content and shape (geometry) of spatial patterns. Integral geometry provides also powerful theorems and formulae, which makes the calculus convenient for many models of stochastic geometrie s, for instance, for the Boolean grain model. This model generates ran dom structures in space by overlapping bodies or ''grains'' (balls, st icks) each with arbitrary location and orientation. We illustrate the integral geometric approach to stochastic geometries by applying morph ological measures to such diverse topics as percolation, complex fluid s, and the large-scale structure of the universe: (A) Porous media may be generated by overlapping holes of arbitrary shape distributed unif ormly in space. The percolation threshold of such porous media can be estimated accurately in terms of the morphology of the distributed por es. (B) Under rather natural assumptions a general expression for the Hamiltonian of complex fluids can be derived that includes energy cont ributions related to the morphology of the spatial domains of homogene ous mesophases. We and that the Euler characteristic in the Hamiltonia n stabilizes a highly connected bicontinuous structure resembling the middle-phase in oil-water microemulsions, for instance. (C) Morphologi cal measures are a novel method for the description of complex spatial structures aiming for relevant order parameters and structure informa tion complement to correlation functions. Typical applications address Turing patterns in chemical reaction diffusion systems, homogeneous p hases evolving during spinodal decomposition, and the distribution of galaxies and clusters of galaxies in the Universe as a prominent examp le of a point process in nature.