Some Mesoscale Issues in Applied Mathematics

当需要一个中间长度或时间尺度来评估系统的总体行为时，或者当不能直接询问更精细的活动尺度时，生物和材料系统中的中尺度现象就会变得突出。 这些系统通常是亚稳态的。 它们为建模、分析和仿真带来了挑战性和新颖的问题。 在这个项目中，我们已经从潜在的应用中直接分离出两个广泛的研究领域。 扩散介导的运输适用于布朗马达和分子棘轮，通常涉及非常小的尺度。在这里，扩散过程和输运过程之间存在协作，扩散过程倾向于通过介质各向同性地传播密度，输运过程倾向于使密度局部化，以产生净输运或功，而单独采取任何一种都不会。从蒙格-康托洛维奇质量输运理论的方法来确定这种类型的系统的亚稳环境，但似乎需要更多的技术，以获得更好的信息。 第二大领域涉及多晶材料的界面，特别是大规模的晶粒生长模拟。 在这里，我们正在实现一种新的数据结构和精心设计的算法，以产生模拟，可以容纳实验得出的能量和流动性的功能，也足够大，以产生可靠的统计。 这是一个基本的问题，实际上得出统计和模拟之间的关系。当需要一个中间长度或时间尺度来评估系统的总体行为时，或者当不能直接询问更精细的活动尺度时，生物和材料系统中的中尺度现象就会变得突出。 这些系统通常是亚稳态的。 它们为建模、分析和仿真带来了挑战性和新颖的问题。 在这里，我们选择了两个具有重要应用的完全不同的领域。 扩散介导的运输位于布朗马达的后面。 最重要的是，这种机制与负责真核细胞运输的马达蛋白有关。 有机会发现化学和力学之间的相互作用，并阐述亚稳态的影响，不能提供一个更令人兴奋的场所。 第二大领域涉及多晶材料中的界面及其大规模模拟。大多数有用的材料是多晶的，由许多被称为晶界的界面分开的小晶粒组成。 这些界面在许多材料特性和许多使用规模中发挥作用。 制备晶粒和边界的排列，即适合于给定目的的织构，是微观结构中的中心问题。实验科学的范式正在发生变化。 自动数据采集技术，现在在材料科学和分子生物学等不同学科中实践，允许在各种尺度范围内进行询问。 这些尺度不一定是最小的，也不一定是最大的，事实上，它们通常是信息丰富的中尺度。 主要的挑战是制定以可靠和有力的方式提取这一信息的战略。 模拟正在成为一个越来越重要的工具，而且，解释这种类型的模拟结果是一个主要问题。我们认为，理解用于询问和建模物理和生物系统的亚稳态系统的大规模模拟的预测特性是计算科学的一个新兴的基本挑战。 当然，这个项目的目标就是迎接这一挑战。

Mesoscale phenomena in biological and material systems assume prominence when an intermediate length or time scale is required to assess gross system behavior or when the finer active scales cannot be directly interrogated. These systems are frequently metastable. They give rise to challenging and novel issues for modeling, analysis, and simulation. In this project we have isolated two broad areas for investigation directly from potential application. Diffusion- mediated transport applies to Brownian motors and molecular ratchets and, typically, involves very small scales. Here, there is a collaboration between a diffusive process, which tends to spread density isotropically through a medium, and a transport process, which tends to localize density, to produce net transport or work when either taken separately would not. Methods from Monge-Kantorovich mass transportation theory are employed to determine the metastable environment of this type of system, but additional techniques seem to be required for better information. The second broad area concerns interfaces in polycrystalline materials and especially the large-scale simulation of grain growth. Here we are implementing a novel data structure and carefully designed algorithms to produce simulations which can accommodate experimentally derived energy and mobility functions and also be large enough to produce reliable statistics. It is a fundamental question to actually derive the relationship between the statistics and the simulation. Mesoscale phenomena in biological and material systems assume prominence when an intermediate length or time scale is required to assess gross system behavior or when the finer active scales cannot be directly interrogated. These systems are frequently metastable. They give rise to challenging and novel issues for modeling, analysis, and simulation. Here, we have chosen two quite different areas with important applications. Diffusion-mediated transport lies behind the Brownian motor. This mechanism is implicated, most importantly, in the motor proteins responsible for eukaryotic cellular traffic. The opportunity to discover the interplay between chemistry and mechanics and to elaborate the implications of metastability could not offer a more exciting venue. The second broad area concerns interfaces in polycrystalline materials and their large-scale simulation. Most useful materials are polycrystalline, comprised of many small grains separated by interfaces called grain boundaries. These interfaces play a role in many material properties and across many scales of use. Preparing arrangements of grains and boundaries, a texture suitable for a given purpose, is a central problem in microstructure. There is a changing paradigm of experimental science. Automated data acquisition technologies, now practiced in disciplines as varied as materials science and molecular biology, allow interrogation at vastly diverse ranges of scales. These scales need not be the smallest nor the largest and, indeed, they are typically those mesoscales which are rich in information. The principal challenge is the development of strategies for the extraction of this information in a reliable and robust way. Simulation is becoming an increasingly important tool and, moreover, interpreting the results of this type of simulation is a major question. We believe that understanding the predictive character of large-scale simulations of metastable systems used to interrogate and model physical and biological systems is an emerging fundamental challenge for computational science. The goal of this project, of course, is to meet this challenge.