Chapter 12
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 pre-
sented for integrals of the product of two and more irreducible tensors. Then the
orientational distribution function needed for orientational averages and the expan-
sion of the distribution with respect to irreducible tensors are introduced, Applica-
tions to the anisotropic dielectric tensor, field-induced orientation of non-spherical
particles, Kerr effect, Cotton-Mouton effect, non-linear susceptibility, the orienta-
tional 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 cor-
relation 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.
12.1 Integrals Over Unit Sphere.
Here, an integral over the unit sphere means a surface integral, as discussed in
Sect.8.2.4.Asin(8.31), d^2 ̂rstands for the surface element of the unit sphere which
is equal to sinθdθdφ, when spherical polar coordinates are used. The surface of
the unit sphere is recalled:
∫
d^2 ̂r= 4 π. Many integrals of interest can be evaluated
effectively, without any explicit integration over angles, by taking the isotropy of
the sphere and the symmetry properties of the integrands into account, and by using
properties of isotropic tensors.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_12
199