Some Mesoscale Issues in Applied Mathematics

当生物和物质系统中的中尺度现象需要一个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，生物和物质系统中的中尺度现象就显得尤为突出。这些系统通常是亚稳定的。它们给建模、分析和仿真带来了具有挑战性的新问题。在这个项目中，我们将两个广泛的领域直接从潜在的应用中分离出来。扩散介导的传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，有一个扩散过程和一个传输过程之间的协作，扩散过程倾向于通过介质各向同性地传播密度，而传输过程倾向于使密度局部化，以产生净传输或功，而两者分开时都不会。Monge-Kantorovich物质输运理论中的方法被用来确定这类系统的亚稳态环境，但为了获得更好的信息，似乎需要额外的技术。第二个广泛的领域涉及多晶材料中的界面，特别是大规模的颗粒生长模拟。在这里，我们正在实施一种新的数据结构和精心设计的算法，以产生模拟，这些模拟可以容纳实验得出的能量和迁移率函数，并且足够大，以产生可靠的统计数据。实际推导统计与模拟之间的关系是一个根本性的问题。当生物和物质系统中的中尺度现象需要一个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，生物和物质系统中的中尺度现象就显得尤为突出。这些系统通常是亚稳定的。它们给建模、分析和仿真带来了具有挑战性的新问题。在这里，我们选择了两个完全不同的具有重要应用的领域。布朗马达背后隐藏着扩散介导的运输。最重要的是，这一机制与负责真核细胞运输的马达蛋白有关。发现化学和力学之间的相互作用和阐述亚稳态的含义的机会是最令人兴奋的场所。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体，由许多被称为晶界的界面分隔的小颗粒组成。这些界面在许多材料属性中发挥作用，并在许多使用范围内发挥作用。制备颗粒和晶界的排列是微观结构中的一个中心问题，这是一种适合特定目的的织构。实验科学的范式正在发生变化。自动化数据采集技术现在应用于材料科学和分子生物学等各种学科，允许在非常不同的范围内进行审讯。这些尺度不一定是最小的，也不一定是最大的，事实上，它们通常是那些信息丰富的中尺度。主要挑战是制定以可靠和有力的方式提取这类信息的战略。模拟正在成为一个日益重要的工具，而且，解释这类模拟的结果是一个主要问题。我们认为，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性是计算科学面临的一个新的基本挑战。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.