Integral Formulae and Distribution Functions

Abstract

This chapter is devoted to integral formulae and distribution functions. Firstly, integrals over the unit sphere are considered, in particular, results are presented for integrals of the product of two and more irreducible tensors. Then the orientational distribution function needed for orientational averages and the expansion of the distribution with respect to irreducible tensors are introduced, Applications to the anisotropic dielectric tensor, field-induced orientation of non-spherical particles, Kerr effect, Cotton-Mouton effect, non-linear susceptibility, the orientational entropy and the Fokker-Planck equation governing the orientational dynamics, are discussed. Secondly, averages over velocity distributions are treated, expansions about a global or a local Maxwell distribution are analyzed and applied for kinetic equations. Thirdly, anisotropic pair correlation functions and static structure factors are considered. Examples for two-particle averages are the potential contributions to the energy and to the pressure tensor of a liquid. The shear-flow induced distortion of the pair-correlation is discussed, in particular for a plane Couette flow. The pair correlation for a system with cubic symmetry is described. The chapter is concluded by a derivation of the quantum-mechanical selection rules for electromagnetic radiation using the expansion of wave functions with respect to irreducible Cartesian tensors.