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.Comment: 17 pages, 2 figure
