Some mesoscale issues for applied mathematics

KinderlehrerDMS-0806703 当需要一个中间长度或时间尺度来评估总的系统行为时，或者当不能直接询问更精细的活动尺度时，物理和生物系统中的中尺度现象就显得突出起来。 这些系统通常是亚稳态的。 两个广泛的研究领域已确定直接从潜在的应用：texturedevelopment依赖于界面性质的多晶和扩散介导的运输与应用合奏的分子马达。 材料科学的一个核心问题是设计一种微结构以获得所需的一系列特性。 研究小组发现了材料织构的新特征，即用几何学和晶体学描述多晶，称为晶界特征分布，发现它与界面能有关。 这个项目的主要目的是解释这种特性采用大规模模拟和分析和随机分析. 几何粗化也进行了研究。 该项目的第二部分的目标是开发适当的建模和分析方法，以了解分子马达在各种情况下的功能，特别是检查各种转化途径和转导方案。 这涉及到非线性分析中的其他新方向的质量输运理论。 理解用于询问和模拟物理和生物系统的亚稳态系统的大规模模拟的预测特征是数学/计算科学的一个新兴的基本挑战。 这是一个粗粒化或升级的问题在信息层面。 本项目的目标就是应对这一挑战。 这个项目有两个相关的部分。 大多数工程材料都是多晶微结构，由称为晶粒的小晶粒的混合物组成，被称为晶界的界面隔开。 这个边界网络的能量学和连通性涉及到从纳米级医学和电子到飞机结构的所有使用尺度的许多属性。 材料织构的一种新的表征方法，即晶界特征分布，是最近由研究小组发现的。 就好像每种材料在显微镜下都留下了独特的足迹。 一个项目的目标是通过模拟和创新的物理分析来解释这种分布，以便为材料工程师提供一个预测工具。 真核细胞内的运输归因于分子马达在细胞骨架或肌动丝上运动的网络。 这是第二部分。 成功地将化学能转化为运动的能力归功于系统中的功能元素和关系。 这些都是建模和分析。有机会发现化学和力学之间的相互作用，并阐述亚稳态的含义，不能提供一个更令人兴奋的场所。

KinderlehrerDMS-0806703 Mesoscale phenomena in physical and biological systemsassume prominence when an intermediate length or time scale isrequired to assess gross system behavior or when the finer activescales cannot be directly interrogated. These systems arefrequently metastable. Two broad areas for investigation havebeen identified directly from potential application: texturedevelopment dependent on interfacial properties of polycrystalsand diffusion mediated transport with application to ensembles ofmolecular motors. A central materials science problem is toengineer a microstructure to attain a desired set ofcharacteristics. The group of the investigator has discovered anew characterization of material texture, the description of apolycrystal in terms of its geometry and crystallography, calledthe grain boundary character distribution, which is found to becorrelated to interfacial energy. A main objective of thisproject is to explain this characterization employing large scalesimulation and analysis and stochastic analysis. Geometriccoarsening is also studied. The goal of the second part of theproject is to develop appropriate modeling and analytical methodsto understand how molecular motors function in variouscircumstances, and in particular, to examine varioustransformation pathways and transduction scenarios. Thisinvolves mass transport theory among other new directions innonlinear analysis. Understanding the predictive character oflarge scale simulations of metastable systems used to interrogateand model physical and biological systems is an emergingfundamental challenge for mathematical/computational science. Itis a coarse graining or upscaling question at the informationallevel. The goal of this project is to address this challenge. This project has two related parts. Most engineeredmaterials arise as polycrystalline microstructures, composed of amyriad of small crystallites, called grains, separated byinterfaces, called grain boundaries. The energetics andconnectivity of this network of boundaries are implicated in manyproperties across all scales of use, from nanoscale medicine andelectronics to aircraft structures. A new characterization ofmaterial texture is now available, the grain boundary characterdistribution, discovered very recently in the group of theinvestigator. It is as if each material leaves a uniquefootprint in the microscope. A project objective is to explainthis distribution by simulation and innovative mathematicalanalysis in order to provide materials engineers with apredictive tool. Eukaryotic intracellular traffic owes to anetwork of molecular motors moving on cytoskeletal or actinfilaments. This is the second part. The ability to successfullytransduce chemical energy to motion owes to functional elementsand relations in the system. These are modeled and analyzed. The opportunity to discover the interplay between chemistry andmechanics and to elaborate the implications of metastabilitycould not offer a more exciting venue.