  We provide a theoretical foundation for the statistical approach for
computing the absorption properties of particles in the Rayleigh domain. We
present a general method based on the Discrete Dipole Approximation (DDA) to
compute the absorption and scattering properties of particles in the Rayleigh
domain. The method allows to separate the geometrical aspects of a particle
from its material properties. Doing the computation of the optical properties
of a particle once, provides them for any set of refractive indices,
wavelengths and orientations. This allows for fast computations of e.g.
absorption spectra of arbitrarily shaped particles. Other practical
applications of the method are in the interpretation of atmospheric and radar
measurements as well as computations of the scattering matrix of small
particles as a function of the scattering angle. In the statistical approach,
the optical properties of irregularly shaped particles are represented by the
average properties of an ensemble of particles with simple shapes. We show that
the absorption cross section of an ensemble of arbitrarily shaped particles
with arbitrary orientations can always be uniquely represented by the average
absorption cross section of an ensemble of spheroidal particles with the same
composition and fixed orientation. This proves for the first time that the
statistical approach is generally viable in the Rayleigh domain.
