Random Matrix Theory for Homogenization of Composites

University Of Utah, Salt Lake City UT

Investigators

Abstract

1715680 Golden 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.