Dynamic constitutive relations for polarization and magnetization.

In this paper we develop constitutive relations for materials where the magnetization and polarization may depend on both the electric and magnetic fields. The approach is general, and is based on a previously developed statistical-mechanical theory. We include the quadrupole-moment density as well as the dipole-moment density in the microscopic displacement field. This yields an electric gradient term in the constitutive equations. This leads to origin invariance in the multipole moments from which Maxwell's equations are defined. We present generalizations of Debye and Landau-Lifshitz equations of motion which are valid for nonequilibrium and contain memory. The reversible and relaxation terms in the polarization and magnetization evolution equations include the possibility of magnetoelectric coupling. Using constitutive relationship, we derive evolution equations for the displacement and induction fields from a Hamiltonian approach.