Some mesoscale issues for applied mathematics

当需要一个中等长度或时间尺度来评估总体系统行为或当更精细的活动尺度不能直接询问时，物理和生物系统中的中尺度现象就显得突出。这些系统经常是亚稳态的。从潜在的应用中直接确定了两个广泛的研究领域：依赖于多晶界面性质的纹理发展和应用于分子马达集成的扩散介导的运输。材料科学的一个核心问题是设计一种微结构以达到所需的一组特性。研究小组发现了材料结构的新特征，从几何和晶体学的角度描述多晶，称为晶界特征分布，它被发现与界面能有关。该项目的一个主要目标是利用大规模模拟和分析以及随机分析来解释这种特征。对几何粗化也进行了研究。该项目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场景。这涉及到质量传递理论和其他非线性分析的新方向。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/计算科学面临的一个新兴的基本挑战。这是一个信息层面上的粗粒度或升级问题。这个项目的目标就是解决这个挑战。这个项目有两个相关的部分。大多数工程材料以多晶微结构的形式出现，由无数的小晶体组成，称为晶粒，由称为晶界的界面分开。这个边界网络的能量学和连通性涉及到所有使用尺度的许多特性，从纳米尺度的医学和电子到飞机结构。现在有了一种新的材料结构表征，即晶界特征分布，这是最近在研究者小组中发现的。就好像每种材料在显微镜下都留下了独特的足迹。该项目的目标是通过模拟和创新的数学分析来解释这种分布，以便为材料工程师提供预测工具。真核生物的细胞内交通是由在细胞骨架或活动丝上移动的分子马达网络构成的。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。对这些进行了建模和分析。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，这是一个最令人兴奋的场所。

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.