Relaxation Processes and Inertial Effects I: Free Rotation about a Fixed Axis

Abstract

The limitations of the usual concepts of relaxation processes for very short periods are discussed, and it is shown that in the limit of very high frequencies the product of the loss factor and frequency must tend to zero. For a consistent kinetic treatment the probability must be considered in phase space, not merely in configuration space; for Cartesian coordinates the stochastic (generalized Liouville) equations can be simplified by a Fourier transformation with regard to the velocities. By means of the simplified equations the dielectric properties are determined for a system of rigid dipoles rotating about fixed axes in a viscous medium. The results are equivalent to, but more compact than, those derived for the same problem by E. P. Gross; for strong viscous damping they agree with Debye's relaxation formulae, except at very high frequencies. The polarization can be described by an infinite discrete set of relaxation times with amplitudes which alternate in sign; the longest relaxation time is identical with that calculated by Debye. The logarithm of the function expressing the dielectric after-effect follows a law similar to the Ornstein-Furth generalization of Einstein's law of diffusion. The analytical solutions obtained by Gross for abrupt collisions between dipoles and heat-bath are also re-derived by a somewhat easier method.