On a Mathematical Framework for the Constitutive Equations of Anisotropic Dielectric Relaxation

Three classes of time-domain non-relativistic anisotropic dielectric constitutive equations of increasing generality are discussed. In each class dissipativity is ensured by the choice of a class of convolution kernels in the D-to-E constitutive equation expressing the electric field E in terms of the electric displacement field D. Defining properties of the inverse (E-to-D) kernels and their Fourier-Laplace transforms (complex dielectric functions) are determined by inversion of the D-to-E constitutive equation. By this procedure it is shown that dielectric functions of the dipolar dielectrics are tensor-valued Bernstein functions while the dielectric functions of the Drude-Lorentz type are tensor-valued negative definite functions. The properties of the complex dielectric permittivities are also determined for either class. The theory is applied to an exhaustive review of empirical response functions of real dielectric materials encountered in the literature. Each class of convolution kernels is consistent with existence of a conserved energy, but in one case a strictly dissipative energy can be constructed.