Some Mesoscale Issues for Applied Mathematics

中尺度是一个用来描述物理系统的中间水平的术语。它的功能是捕捉系统在更细尺度上与更大的粗尺度和中尺度本身的相互作用或影响的相互作用。这样的系统，在不同的长度和时间尺度上活动，本质上是亚稳态的。这种特征通常表现为滞后行为或不愿迅速进化到平衡状态。本提案的重点是在材料科学中出现的这些系统的几个原型。一个是界面或晶界在决定或限制多晶材料行为中的作用。根据既定的热力学原理，晶界的能量和迁移率取决于晶体学和几何结构。创新的新方法明确地确定这些重要材料的功能是中尺度界面映射项目的目标。这包括开发自动显微镜从样品中收集大量数据，然后制定和解决一个复杂的反问题。确定迁移率的一种方法是发展大规模的晶界演化模拟。第二个重点是中尺度系统的粗粒度描述，固体微观结构中的功能解析极限过程或随机行为系统中的平均到分布函数。新方法允许研究动力学直接出现在热力学状态函数方面的情况，并且自然地带有适当的拓扑结构。在这种背景下，亚稳态的问题一直在研究中。现在有机会提高对形状记忆材料的微观结构演变的理解，以及某些液晶系统和蛋白质马达中扩散介导的运输。这将包括对这些系统的诊断。材料科学中尺度的挑战是理解它如何约束更精细的系统（在分子尺度上）和决定更大的系统（在整个设备的尺度上）。这是通过粗粒化过程实现的。例如，许多技术上有用的材料本质上是多晶的或颗粒状的。飞机的铝外壳和计算机芯片中的铜或铜铝互连只是这类颗粒材料在大小尺度上截然不同的两个例子。人们普遍认为，这些材料的许多方面取决于它们所含的界面或它们的晶界。晶界的性质决定了材料的可靠性和机械强度。该项目将为这一领域及其他领域的数学科学探索令人兴奋的机遇和挑战。具体地说，粗粒化方法将得到发展，以便更好地理解发生在蛋白质马达和液晶中的现象。当要从复杂系统（如多晶材料）的模拟产生的大量数据中提取信息时，就会出现粗粒度的相关问题。这个在信息尺度而不是物理尺度上的粗粒度问题也将在这个项目中得到解决。

Mesoscale is a term intended to convey an intermediate level description ofa physical system. Its function is to capture the interactions of thesystem at finer scales with interactions or influences from larger coarserscales, and the mesoscale level itself. Such systems, active acrossdisparate length and time scales, are inherently metastable. This featureis often revealed by hysteretic behavior or by a reluctance to evolvequickly to equilibrium. The focus in this proposal is on severalprototypes of these systems that occur in materials sceince. One is the role of interfaces, or grain boundaries, in determining or limiting the behavior of polycrystalline materials. The energy and mobility of grain boundaries depends on crystallography and geometry, according to established thermodynamic principles. Innovative new ways to determine these functions explicitly for important materials are the objective of the Mesoscale Interface Mapping Project. This involves developing automated microscopy to harvest large amounts of data from samples and then formulating and solving a complex inverse problem. One way to approach determination of mobility consists in the development of large scale simulations of grain boundary evolution. The second focus is the coarse grained descriptions of mesoscale systems, the functional analytic limit processes in the microstructure of solids or the averaging to distribution functions in systems with stochastic behavior. The new methods allow study of situations where kinetics arise directly in terms of thermodynamic state functions and naturally carry with them an appropriate topology. The issue of metastability has been under investigation in this context. There is now the opportunity to improve understanding, for example, of microstructural evolution in shape-memory materials and diffusion mediated transport in certain liquid crystal systems and in protein motors. This will include diagnostics for these systems.The challenge of the mesoscale in materials science is to understand how it constrains finer scale systems (at the molecular scale) and determines larger scale systems (at the scale of entire devices). This is accomplishedthrough coarse graining procedures. For example, many technologicallyuseful materials are polycrystalline, or granular, in nature. The aluminumskin of an aircraft and the copper or copper-aluminum interconnects incomputer chips are but two examples at vastly different size scales of suchgranular materials. It is widely understood that many aspects of thesematerials depend on the interfaces they contain, or their grain boundaries.Properties of grain boundaries determine the reliability as well as the mechanical strength. This project will exploit the exciting opportunities and challenges for mathematical science in this field and beyond. Specifically, coarse graining methods will be developed in order to better understand phenomena that occur in protein motors and in liquid crystals. A related problem of coarse graining arises when information is to be extracted from the immense amounts of data that can be produced with simulations ofcomplex systems, such as polycrystalline materials. This problem of coarse graining at an information scale rather than a physical scale will also be addressed in this project.