Relaxation Processes and Inertial Effects II: Free Rotation in Space

Abstract

The calculations of the effect of inertia on the dielectric properties of systems of identical non-interacting rigid dipoles with various types of fluctuation are extended to the case of free rotation in space. The equations of motion, and hence the stochastic equations, depend on all three moments of inertia; the dumb-bell and the spherical top models are treated in detail. The probability distribution in phase space is conveniently replaced by a distribution in configuration-velocity space which simplifies the fluctuation terms, but necessitates a modification of the Liouville equation. The response of the polarization to an alternating electric field can be expressed in terms of continued fractions similar to the solution obtained by E. P. Gross for the plane rotation model. For all collision mechanisms which leave the orientation of the dipoles unaltered the second approximation yields a modification of Debye's relaxation formula which has previously been derived by Powles and by Gross; the precise nature of the model affects the result only in higher approximation. The relation of the exact solutions and several approximate formulae is discussed and a misprint in a formula by Rocard is pointed out. For fluctuations of the Browman movement type the frequency response of the dumb-bell model tends towards that of a damped harmonic oscillator as the viscosity decreases. For abrupt collisions the results can be expressed in an analytic form.