Duality and exact results for conductivity of 2D isotropic heterophase systems in magnetic field

TL;DR

This paper derives exact formulas for the effective conductivity of 2D isotropic heterophase systems in magnetic fields using group transformations, revealing conditions where conductivity is independent of phase concentrations.

Contribution

It introduces a group-theoretic method to find exact effective conductivities in heterophase systems under magnetic fields, extending previous binary system results to more complex cases.

Findings

Exact effective conductivity values are obtained for specific hypersurfaces in partial conductivity space.

Effective conductivity in certain cases is independent of phase concentrations.

A 3-parametric transformation relates conductivities with and without magnetic fields.

Abstract

Using a fact that the effective conductivity sigma_{e} of 2D random heterophase systems in the orthogonal magnetic field is transformed under some subgroup of the linear fractional group, connected with a group of linear transformations of two conserved currents, the exact values for sigma_{e} of isotropic heterophase systems are found. As known, for binary (N=2) systems a determination of exact values of both conductivities (diagonal sigma_{ed} and transverse Hall sigma_{et}) is possible only at equal phase concentrations and arbitrary values of partial conductivities. For heterophase (N > 2) systems this method gives exact values of effective conductivities, when their partial conductivities belong to some hypersurfaces in the space of these partial conductivities and the phase concentrations are pairwise equal. In all these cases sigma_e does not depend on phase concentrations. The complete, 3-parametric, explicit transformation, connecting sigma_e in binary systems with a magnetic field and without it, is constructed