Linear response of general composite systems to many coupled fields

Abstract

We treat the linear response of an arbitrary composite system to many coupled driving fields, as in the thermoelectric or magnetoelectric effects. The system is made of any number of, possibly anisotropic, components. The response consists of some component of the fluxes (currents) measured at an arbitrary point in the system and is considered as responding to the boundary conditions. We expound the problem pertaining to such a response and investigate the general properties of the relevant response matrices. Let ${\mathit{L}}_{\mathit{k}\mathit{m}}^{\mathit{a}}$ be the elements of the p response matrices, ${\mathit{L}}_{\mathit{a}}$ (1≤a≤p), characterizing the components. We show that the functional dependence of the response coefficients (${\mathit{scrL}}_{\mathit{i}\mathit{j}}$) of the composite on them [${\mathit{scrL}}_{\mathit{i}\mathit{j}}$=${\mathit{scrL}}_{\mathit{i}\mathit{j}}$(${\mathit{L}}_{\mathit{k}\mathit{m}}$ ${}_{}{}^{\mathit{a}}{}_{}{}^{}$)] is subject to two constraints: First, it enjoys certain covariance properties under linear transformations of the fields; for any real regular matrix W we must have ${\mathit{W}}^{\mathrm{-}1}$scrL(${\mathit{WL}}_1$W̃,...,${\mathit{WL}}_{\mathit{p}}$W̃)W${\mathrm{̃}}^{\mathrm{-}1}$=s crL(${\mathit{L}}_1$,...,${\mathit{L}}_{\mathit{p}}$), where W̃ is the transpose of W.