On the derivation of the Debye theory of dielectric relaxation from the Langevin equation in the presence of the driving field. II. Inclusion of inertial effects for rotation in three dimensions

Abstract

It is shown how the inertial response in the presence of a driving field of an assembly of molecules free to rotate in space (with dipole–dipole coupling effects ignored) may be obtained by direct averaging of the Langevin equation. The procedure is illustrated by considering the linear dielectric response of an assembly of needle like rotators choosing as variables in the Langevin equation the Hermite polynomials in the angular velocities and the associated Legendre functions of order l. This leads directly, using the statistical properties of the white noise driving torques, to the differential–difference equations which govern the dielectric response. These are converted into an infinite set of linear simultaneous first order differential equations which are solved by successively limiting the size of the transition matrix to 2×2, 3×3, etc. The continued fraction solution previously obtained by Sack is recovered using this procedure. The 2×2 approximation to the solution, effectively corresponding to the Rocard equation, is discussed in relation to recent far infrared absorption measurements.