Kramers-Kronig relations and the properties of conductivity and permittivity in heterogeneous media

TL;DR

This paper demonstrates that the frequency-dependent electric conductivity and permittivity in heterogeneous media are interconnected through Kramers-Kronig relations, imposing important physical constraints on their models and measurements.

Contribution

It extends the Kramers-Kronig relations to electric conductivity, providing a new theoretical framework for analyzing heterogeneous media properties.

Findings

Kramers-Kronig relations apply to conductivity, not just permittivity.

Constraints ensure physically plausible frequency dependence of electric properties.

Examples illustrate the application of these relations to real media.

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 exists 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.