Some mesoscale issues for applied mathematics

KinderlehrerDMS-0806703物理和生物系统中的中尺度现象在需要中等长度或时间尺度来评估总体系统行为时，或者当不能直接询问更精细的活动尺度时，表现得尤为突出。这些系统通常是亚稳定的。直接从潜在的应用中确定了两个广泛的研究领域：依赖于多晶界面性质的织构发展和扩散介导的传输，并应用于分子马达的系综。材料科学的一个核心问题是设计一种微结构，以获得所需的一组特性。研究小组发现了材料织构的一种新的表征，即从几何和结晶学的角度描述多晶，称之为晶界特征分布，发现它与界面能有关。这个项目的一个主要目标是用大规模的模拟和分析以及随机分析来解释这种特征。并对几何粗化进行了研究。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如何发挥作用，特别是检查各种转化途径和转导情景。这涉及到质量输运理论和其他非线性分析的新方向。对于数学/计算科学来说，理解用于询问和模拟物理和生物系统的大规模亚稳态系统模拟的预测特性是一个新出现的基本挑战。这是一个信息层面上的粗粒化或升级问题。该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以多晶微结构的形式出现的，这种结构由许多称为颗粒的小微晶组成，由称为晶界的界面隔开。这种边界网络的能量学和连通性涉及到所有使用范围的许多特性，从纳米级医学和电子学到飞机结构。材料织构的一种新的表征方法现已问世，晶界特征分布是该研究小组最近发现的。就好像每种材料都在显微镜上留下了独一无二的足迹。一个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。真核细胞内交通是由于分子马达在细胞骨架或放线丝上运动的网络所致。这是第二部分。将化学能成功地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。发现化学和力学之间的相互作用和阐述亚稳定的含义的机会是最令人兴奋的场所。

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.