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

Using a fact that the effective conductivity σˆe 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 σˆe of isotropic heterophase systems are found. As known, for binary (N=2) systems a determination of exact values of both conductivities (diagonal σed and transverse Hall σet) is possible only at equal phase concentrations and arbitrary values of partial conductivities. For heterophase (N⩾3) 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 some phase concentrations are pairwise equal. In all these cases σe does not depend on phase concentrations. The complete, 3-parametric, explicit transformation, connecting σe in binary systems with a magnetic field and without it, is constructed.