A RELATIONSHIP BETWEEN ASYMMETRIC LEVY-STABLE DISTRIBUTIONS AND THE DIELECTRIC SUSCEPTIBILITY

This paper, as a complement to the work of Montroll and Bendler, is concerned with the Levy-stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain. The necessary and sufficient condition for this connection is found. The presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates, namely, the completely asymmetric Levy-stable distribution, should apply in all different relaxing systems. This is in contrast to the traditional definition of the relaxation function, expressed as a weighted average of exponential relaxation functions, which does not explain the universality of the dielectric relaxation law. It also follows from the present considerations that not only is the imaginary part chi''(omega) of the dielectric susceptibility directly related to the Levy-stable distribution (as was found by Montroll and Bendler), but so is the real part chi'(omega). As a consequence the relation chi''(omega)/chi'(omega) = cot(npi/2) for omega > omega(p) and 0 < n < 1, implied by experimental results, is obtained.