Random Matrix Theory for Homogenization of Composites

小行星1715680 复合材料因其独特的物理和化学性能上级传统产品而在工程和制造业中受到高度重视。 科学和工业界一直在寻找比传统复合材料更坚固、更轻、更便宜的复合材料。 这些结构化介质-或超材料-可能会表现出更高的导电性，增强的隔热或隔音性能，更可靠的耐用性，甚至似乎无法实现的特性，如不可见性。 具体的例子包括用于飞机的更轻的机翼和机身材料，用于军事的防弹衣，用于整形外科手术的人工关节和专业运动设备。 复合材料也出现在整个自然界-在人类和动物的身体，并在地球内部和表面上发现的大多数组件。 例子包括骨头、肺、含有石油和天然气的多孔岩石、农业土壤和海冰。 复合材料的均匀化数学理论已经发展到解释现有复合材料的观察到的有效性能，然后可以用来预测和发现新的复合材料，而不需要昂贵的实验。 最近，研究人员发现了复合材料的均匀化与安德森的金属-绝缘体转变理论之间的一个意想不到的数学相似之处，安德森因此分享了诺贝尔奖。 在这个项目中，研究人员开发了新的方法来研究复合材料的基础上，这种平行，使强大的思想的安德森转变承担在复合材料理论的广泛问题。 研究生和本科生参与该项目的工作。 本项目通过随机矩阵理论的透镜研究了复合介质中输运的均匀化谱理论。 均匀化问题的一个强大的方法是解析延拓方法，它通过一个自伴随机算子的频谱测量来编码有关复合材料微观结构的信息，该算子控制介质中的经典输运。 随机矩阵理论自然产生考虑有限离散模型的复合材料，这使得频谱测量，从而宏观行为的复合材料，计算的特征向量和特征值的随机矩阵。 令人惊讶的是，当接近渗流阈值时，这些本征值和本征向量显示出与在凝聚态、光学、声学和水波中的安德森跃迁中观察到的行为惊人地相似。 这种意想不到的联系使研究人员能够开发新的分析和计算方法，用于两相复合材料和相关系统（如多晶和对流扩散过程）的均匀化。 此外，他们的方法将以前不相关的随机矩阵理论和均匀化领域联系在一起，为研究和应用开辟了新的途径。 研究生和本科生参与该项目的工作。

1715680Golden Composite materials are highly valued in engineering and manufacturing for their unique physical and chemical properties that are superior to those of conventional products. Science and industry are continually searching for composites that are stronger, lighter, and less expensive than their traditional counterparts. These structured media -- or metamaterials -- may exhibit increased electrical conductivity, enhanced thermal or acoustic insulation properties, more reliable durability, or even seemingly unattainable properties such as invisibility. Specific examples include lighter wings and fuselage materials for aircraft, body armor for the military, artificial joints used in orthopedic surgery, and professional sporting equipment. Composites also appear throughout the natural world -- in human and animal bodies, and in most components found within and on the surface of the Earth. Examples include bone, lungs, porous rocks containing oil and gas, agricultural soils, and sea ice. The mathematical theory of homogenization for composite materials has been developed to explain observed effective properties of existing composites, which can then be used to predict and discover new composites with less need for costly experimentation. Recently the investigators discovered an unexpected mathematical parallel between homogenization for composites and the Anderson theory of the metal-insulator transition, for which Anderson shared the Nobel prize. In this project the investigators develop new methods for studying composites based on this parallel, bringing the powerful ideas of the Anderson transition to bear on a broad range of problems in the theory of composites. Graduate and undergraduate students participate in the work of the project. In this project the spectral theory of homogenization for transport in composite media is investigated through the lens of random matrix theory. A powerful approach to homogenization problems is the analytic continuation method, which encodes information about the microstructure of the composite through the spectral measure of a self-adjoint random operator governing classical transport in the medium. Random matrix theory naturally arises by considering finite discrete models of composites, which allows the spectral measure, and thus the macroscopic behavior of the composite, to be computed in terms of the eigenvectors and eigenvalues of the random matrices. Surprisingly, as a percolation threshold is approached, these eigenvalues and eigenvectors display strikingly similar behavior to what is observed in Anderson transitions in condensed matter, optics, acoustics, and water waves. This unexpected connection enables the investigators to develop new methods of analysis and computation for homogenization of two-phase composites and related systems such as polycrystals and advection-diffusion processes. Moreover, their approach ties together previously unrelated fields of random matrix theory and homogenization, opening up new avenues for investigation and application. Graduate and undergraduate students participate in the work of the project.