各向异性分数阶麦克斯韦-科尔-科尔介质中的能量、无源性与扩散表示
Energy, Passivity, and Diffusive Representations in Anisotropic Fractional Maxwell--Cole--Cole Media
AI总结：
本研究构建各向异性分数阶麦克斯韦-科尔-科尔介质的能量公式，推导全局恒等式确立无源性与能量衰减，恢复多种极限，验证扩散表示，揭示分数阶对介电特性的影响，关联多种电磁特性。
摘要：
本研究针对具有多个分数阶极化通道、无源电导率、规定电流以及零预历史的有限终端Caputo记忆的线性各向异性麦克斯韦-科尔-科尔介质，构建了能量公式。加权解类捕获由静止极化产生的可积分数阶初始层。分数阶极化功率随后分解为非负的历史存储和非负的本构耗散率。尽管瞬时率在初始化时可能发散，但它仍保持可积性，而存储的历史能量和累积损耗在初始时消失。结合坡印廷定理，该分解得到全局场-记忆恒等式，其在零供应边界条件下确立了电磁无源性、封闭系统能量衰减和条件唯一性。该框架在收敛假设下还恢复了德拜、瞬时介质、零极化率和真空极限。正扩散表示提供了第二种分解，其存储量从不超过历史存储量，二者之差为精确的非负加权方差，仅在极化历史为零时消失。与这些时域结果一致，有限时间谐波响应趋近于各向异性多通道科尔-科尔介电常数，其厄米虚部确保非负的周期平均吸收。一项可重复的单通道研究表明，分数阶会拓宽介电损耗带、重新分配波衰减、增强可积初始功率层并延长阶跃响应尾。精确的电阶跃响应验证了两种能量表示，该公式将分数阶初始层、遗传存储、无源性、扩散弛豫、色散、吸收和衰减关联到局部本构模型中。
英文摘要：
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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