Abstract
The aim of this paper is to point out the importance of geometric functionals in statistical physics. Integral geometry furnishes a suitable family of morphological descriptors, known as Minkowski functionals, which are related to curvature integrals and do not only characterize connectivity (topology) but also content and shape (geometry) of spatial patterns. Since many physical phenomena depend essentially on the geometry of spatial structures, integral geometry may provide useful tools to study physical systems, in particular, in combination with the Boolean model, well known in stochastic geometry. This model generates random structures by overlapping ‘grains’ (spheres, sticks) each with arbitrary location and orientation. The integral geometric approach to stochastic structures in physics is illustrated by applying morphological measures to such diverse topics as complex fluids, porous media and pattern formation in dissipative systems. It is not intended to cover these topics completely but to emphasize unsolved physical problems related to geometric features and to present ideas and proposals for future work in possible collaboration with spatial statisticians and statistical physicists.
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