Exact moments of curvature measures in the Boolean model - Mecke, Klaus (Journal of Statistical Physics, 2001)
Curvature measures are important for the characterization of spatial structures since many physical phenomena depend essentially on the geometry of spatial configurations. Curvature-weighted correlation functions can be defined and calculated explicitly within the Boolean model. This standard model in statistical physics generates random geometries by overlapping grains (spheres, sticks) each with arbitrary location and orientation. A general decomposition relation for characteristic functions of submanifolds based on Chern's differential kinematic formula is used to obtain exact expressions for mean values and second order moments of curvature integrals, i.e., of Minkowski functionals. Am exact relation for the canonical and grand-canonical second-order moments of the morphological Minkowski measures can be derived based only on thermodynamic arguments. The results are applied on the morphological thermodynamics of complex fluids where specific heats are expressed in terms of curvature moments.
Keywords: random sets; integral geometry; Minkowski functionals; Euler characteristic; Boolean model