Integral Geometry in Statistical Physics

Abstract

The aim of this paper is to point out the importance of a morphological characterization of patterns in Statistical Physics. Integral geometry furnishes a suitable family of morphological descriptors, known as Minkowski functionals. They characterize not only the connectivity (topology) 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 geometries, for instance, for the Boolean grain model. This model generates random structures in space by overlapping bodies or "grains" (balls, sticks) each with arbitrary location and orientation. We illustrate the integral geometric approach to stochastic geometries by applying morphological measures to such diverse topics as percolation, complex fluids, and the large-scale structure of the universe: (A) Porous media may be generated by overlapping holes of arbitrary shape distributed uniformly in space. The percolation thresold of such porous media can be estimated accurately in terms of the morphology of the distributed pores. (B) Under rather natural assumptions a general expression for the Hamiltonian of complex fluids can be derived that includes energy contributions related to the morphology of the spatial domains of homogeneous mesophases. We find that the Euler characteristic in the Hamiltonian stabilizes a highly connected bicontinuous structure resembling the middle-phase in oil-water microemulsions, for instance. (C) Morphological measures are a novel method for the description of complex spatial structures aiming for relevant order parameters and structure information complement to correlation functions. Typical applications address Turing patterns in chemical reaction diffusion systems, homogeneous phases evolving during spinodal decomposition, and the distribution of galaxies and clusters of galaxies in the Universe as a prominent example of a point process in nature.