Analytical and numerical analysis of Debye–Ramm’s equation for the polarization density distribution in polar materials

Abstract

Due to dipole interaction, the molecular polarization brought about by an external field is significantly lower in condensed matter (liquids) than in a gas. In addition to this, the response of interacting dipoles to stepwise changes of the external field does not follow a simple time exponential. Instead, a spectrum of relaxation times is required to describe such a response. Debye and Ramm [Ann. Phys. 28, 28 (1937)] have attempted to describe the effects associated with rotational hindrances due to dipole-dipole interaction by the following differential equation: ∂f/∂t =(kT/ρ)Δf+(1/ρ)div(f grad u), where f denotes the distribution function specifying the number of dipoles pointing in a certain solid angle, t the time, ρ a friction coefficient, and u the potential of the forces acting on the dipoles. The latter quantity depends both on the external field and on the contribution from the dipole-dipole interaction (internal field). Although unable to solve the above equation explicitly, Debye and Ramm (DR) made some predictions about the solution, concluding, among other things, that the inclusion of an internal field E would yield a process with a discrete spectrum of relaxation times. Finding such prospect highly interesting, we subjected the DR equation to a close study using some advanced mathematical tools (Fourier integral operators etc.). Contrary to the conclusions of DR, we found that the above equation cannot be solved in the way originally described, and that the conjectured eigenfunctions and eigenvalues do not exist. Furthermore, we show that, in contrast to DR’s statements, the above equation is not solved by certain classical expressions relating to free-rotating dipoles (no internal field). The lack of physical content of this equation appears to be due to a number of not permissible simplifications.