TITLE:
Kramers-Kronig Relations and the Properties of Conductivity and Permittivity in Heterogeneous Media
AUTHORS:
Claude Bédard, Alain Destexhe
KEYWORDS:
Inhomogeneous Materials, Impedance, Mean-Field, Maxwell Equations, Hilbert Transform
JOURNAL NAME:
Journal of Electromagnetic Analysis and Applications,
Vol.10 No.2,
February
13,
2018
ABSTRACT: The macroscopic electric permittivity of a given medium may depend on frequency, but this frequency dependence cannot be arbitrary, its real and imaginary parts are related by the well-known Kramers-Kronig relations. Here, we show that an analogous paradigm applies to the macroscopic electric conductivity. If the causality principle is taken into account, there exist Kramers-Kronig relations for conductivity, which are mathematically equivalent to the Hilbert transform. These relations impose strong constraints that models of heterogeneous media should satisfy to have a physically plausible frequency dependence of the conductivity and permittivity. We illustrate these relations and constraints by a few examples of known physical media. These extended relations constitute important constraints to test the consistency of past and future experimental measurements of the electric properties of heterogeneous media.