Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems

In a variety of three-dimensional, multi-particle systems, interactions of tensorial form occur between individual components and an applied stimulus that operates uniformly throughout the ensemble. When each material component has an identical, fixed orientation, its own response is replicated in the observed form of behaviour by the system as a whole, with respect to the angular disposition of the stimulus. The same complete correlation between microscopic and macroscopic response does not, however, operate in other systems where there is a degree of orientational freedom amongst the microscopic components. One limiting case is where the extent of such freedom allows random orientations, such conditions delivering a system that is macroscopically isotropic. When each particle has incomplete orientational freedom, the formulation is in general less mathematically tractable; here, the theory required to describe ensemble response yields analytically solvable equations only in cases where the distribution function takes a relatively simple form, such as a scalar product, and few explicit results are available. This paper addresses general systems in which there is an orientation-dependent complex exponential factor, weighting the response of each component. By means of irreducible tensor decomposition, results are determined for tensor interactions up to and including rank 2, where the weighting exponent is also of any rank up to the same order. Illustrative applications are drawn from the theory of laser particle orientation and photon absorption.