ANALYTICAL AND NUMERICAL-ANALYSIS OF DEBYE-RAMMS EQUATION FOR THE POLARIZATION DENSITY DISTRIBUTION IN POLAR MATERIALS

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 dipo les 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 hi ndrances due to dipole-dipole interaction by the following differentia l equation: delta f/delta f=(kT/rho)Delta f+(1/p)div(f grad u), where f denotes the distribution function specifying the number of dipoles p ointing in a certain solid angle, t the time, rho a friction coefficie nt, and u the potential of the forces acting on the dipoles. The latte r quantity depends both on the external field and on the contribution from the dipole-dipole interaction (internal field). Although unable t o solve the above equation explicitly, Debye and Ramm (DR) made some p redictions about the solution, concluding, among other things, that th e inclusion of an internal field E would yield a process with a discre te spectrum of relaxation times. Finding such prospect highly interest ing, 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 eigenfunctio ns and eigenvalues do not exist. Furthermore, we show that, in contras t to DR's statements, the above equation is not solved by certain clas sical 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.