Some Mesoscale Issues in Applied Mathematics

当需要一个中间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，生物和材料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。它们为建模、分析和仿真带来了具有挑战性和新颖的问题。在这个项目中，我们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。这里，在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向各向同性传播密度，输运过程倾向于局部化密度，产生净输运或功，而单独采取任何一种都不会。Monge-Kantorovich质量输运理论的方法被用来确定这类系统的亚稳态环境，但似乎需要额外的技术来获得更好的信息。第二个广泛的领域涉及多晶材料中的界面，特别是晶粒生长的大规模模拟。在这里，我们正在实现一种新颖的数据结构和精心设计的算法，以产生可以容纳实验导出的能量和迁移函数的模拟，并且也足够大以产生可靠的统计数据。推导统计和模拟之间的关系是一个基本问题。当需要一个中间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，生物和材料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。它们为建模、分析和仿真带来了具有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。布朗马达的背后是扩散介导的输运。最重要的是，这种机制与负责真核细胞运输的运动蛋白有关。有机会发现化学和力学之间的相互作用，并阐述亚稳态的含义，这是一个最令人兴奋的场所。第二个广泛的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，由称为晶界的界面分隔的许多小晶粒组成。这些界面在许多材料特性和许多使用尺度中发挥作用。制备晶粒和边界的排列，适合于给定目的的织构，是微观结构的中心问题。实验科学的范式正在发生变化。自动化数据采集技术现已在材料科学和分子生物学等各种学科中得到应用，它允许在各种规模的范围内进行讯问。这些尺度不一定是最小的，也不一定是最大的，实际上，它们通常是信息丰富的中尺度。主要的挑战是开发以可靠和健壮的方式提取这些信息的策略。模拟正成为一种越来越重要的工具，而且，解释这种模拟的结果是一个主要问题。我们相信，理解亚稳态系统的大规模模拟的预测特征，用于询问和模拟物理和生物系统，是计算科学面临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.