麦克斯韦方程组平均的基础
Basics of averaging of the Maxwell equations
摘要：
微观麦克斯韦方程组的体积平均或统计平均，即从微观麦克斯韦方程组到宏观麦克斯韦方程组的过渡，是材料电动力学研究的主要步骤之一。尽管求平均值的过程具有根本的重要性，但在大学的课程和相应的书籍中很少有适当的讨论;到目前为止，关于如何进行求平均值的过程还没有达成共识。在本文中，我们表明，有一些基本原则的平均程序（无论什么类型的材料进行研究），必须满足。任何均匀化模型都必须符合基本原则。如果某一特定模式与基本原则之间缺乏这种关联，则该模式不能被视为可信的模式。本文的另一个目标是建立超材料的平均程序，这是相当接近的情况下，复合材料，但应包括磁性响应的夹杂物和它们的集群。我们从考虑体材料开始，这意味着在绝大多数情况下，我们考虑远离界面的电磁波的传播，其中介质中的本征波已经形成并稳定。关于边界条件和分层超材料的讨论的基本结构是一个单独出版的主题，将在其他地方进行。
英文摘要：
Volume or statistical averaging of the microscopic Maxwell equations (MEs), i.e. transition from microscopic MEs to their macroscopic counterparts, is one of the main steps in electrodynamics of materials. In spite of the fundamental importance of the averaging procedure, it is quite rarely properly discussed in university courses and respective books; up to now there is no established consensus about how the averaging procedure has to be performed. In this paper we show that there are some basic principles for the averaging procedure (irrespective to what type of material is studied) which have to be satisfied. Any homogenization model has to be consistent with the basic principles. In case of absence of this correlation of a particular model with the basic principles the model could not be accepted as a credible one. Another goal of this paper is to establish the averaging procedure for metamaterials, which is rather close to the case of compound materials but should include magnetic response of the inclusions and their clusters. We start from the consideration of bulk materials, which means in the vast majority of cases that we consider propagation of an electromagnetic wave far from the interfaces, where the eigenwave in the medium has been already formed and stabilized. A basic structure for discussion about boundary conditions and layered metamaterials is a subject of separate publication and will be done elsewhere.
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