Keller's Theorem Revisited

TL;DR

This paper revisits Keller's theorem, providing a new derivation applicable to dispersive and dissipative anisotropic systems, and clarifies its validity at zero frequency through numerical verification.

Contribution

A novel derivation of Keller's theorem that does not rely on common assumptions and extends its applicability to complex anisotropic systems.

Findings

The theorem is strictly valid only at zero frequency.

Numerical verification confirms the theorem in various systems.

Discusses implications for dielectric response analysis.

Abstract

Keller's theorem relates the components of the macroscopic dielectric response of a binary two-dimensional composite system with those of the reciprocal system obtained by interchanging its components. We present a derivation of the theorem that, unlike previous ones, does not employ the common asumption that the response function relates an irrotational to a solenoidal field and that is valid for dispersive and dissipative anisotropic systems. We show that the usual statement of Keller's theorem in terms of the conductivity is strictly valid only at zero frequency. We verify the theorem numerically in several ordered and disordered systems and discuss some of its consequences.