Energy, Passivity, and Diffusive Representations in Anisotropic Fractional Maxwell--Cole--Cole Media

Abstract

This study develops an energy formulation for linear anisotropic Maxwell--Cole--Cole media with multiple fractional polarization channels, passive conductivity, prescribed currents, and finite-terminal Caputo memory with zero prehistory. A weighted solution class captures the integrable fractional initial layer generated by quiescent polarization. Fractional polarization power then separates into a nonnegative history storage and a nonnegative constitutive dissipation rate. Although the instantaneous rate may diverge at initialization, it remains integrable, while stored history energy and accumulated loss vanish initially. Combined with the Poynting theorem, this decomposition yields a global field-plus-memory identity that establishes electromagnetic passivity, closed-system energy decay, and conditional uniqueness under zero-supply boundary conditions. The framework also recovers the Debye, instantaneous-dielectric, vanishing-susceptibility, and vacuum limits under convergence assumptions. A positive diffusive representation provides a second decomposition whose storage never exceeds the history storage. Their difference is an exact nonnegative weighted variance that vanishes only for a zero polarization history. In agreement with these time-domain results, the finite-time harmonic response approaches the anisotropic multichannel Cole--Cole permittivity, whose Hermitian imaginary part ensures nonnegative cycle-averaged absorption. A reproducible single-channel study shows that fractional order broadens the dielectric-loss band, redistributes wave attenuation, strengthens the integrable initial power layer, and prolongs the step-response tail. An exact electric-step response verifies both energy representations. The formulation connects fractional initial layers, hereditary storage, passivity, diffusive relaxation, dispersion, absorption, and attenuation in a local constitutive model.

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