We characterize the fluctuations in disordered systems of overlapping grains by calculating the second moments of a quite general class of robust shape descriptors. These include volume, surface area, Euler characteristic, and all other Minkowski functionals and tensors.
D. Hug, M. A. Klatt, G. Last, M. Schulte. Second order analysis of geometric functionals of Boolean models. In Lecture Notes in Mathematics Tensor Valuations and their Applications in Stochastic Geometry and Imaging (eds. E. B. Vedel Jensen and M. Kiderlen) 2177:339–383 (Springer International Publishing, 2017)
We especially focus on models of anisotropic heterogenous materials, that is, on anisotropic grain distributions.
How does the behaviour of the second moments and probability distributions differ for various models in finite systems for specific examples like aligned or isotropically
oriented rectangles? We derive explicit formulae, which are compared to the results of simulations. We show analytically that the asymptotic formulae for the thermodynamic limit are actually exact even for finite systems with periodic boundary conditions.
In the thermodynamic limit, the geometric functionals follow normal distributions, where the first and second moments are (explicitly) known, which could be used to construct hypothesis tests for random heterogeneous media.