Some Mesoscale Issues in Applied Mathematics

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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.