FRG: Fluctuation Effects in Near-Continuum Descriptions of Discrete Dynamical Systems in Physics, Chemistry and Biology

这个重点研究小组汇集了密歇根大学数学、物理和化学工程系的研究人员，以解决动力学过程的建模、模拟和分析的重要问题，其中潜在的离散性在通过确定性连续统系统（通常是常微分方程和偏微分方程系统）的大规模描述中起着不可忽略的作用。该研究小组结合了研究人员在理论、建模、分析和科学计算方面的专业知识，从材料物理学、化学动力学和生命科学等方面研究一系列问题，以阐明基本的科学问题，并开发适当的定量工具来分析它们。需要研究的具体问题有：(1)伤口愈合与细胞增殖和迁移的介观数学模型，包括细胞-细胞粘附的重要生物学效应；(2)应用新的和改进的模拟技术，直接求解Becker-Doering方程，模拟和分析随机模型，研究微观相关性在奥斯特瓦尔德成熟中的作用；(3)发展了分析渐近方法，以准确地描述慢随机变量，适当地结合残余波动效应，并将其应用于具有宽反应速率谱的（生物）化学反应网络；(4)将简单系统的建模、分析和模拟方法扩展到种群生物学和流行病学中日益复杂的随机模型，包括结构种群中的流行病和竞争物种的灭绝；(5)随机Fisher-Kolmogorov方程的空间非均质性和反应速率变化，这是锋面传播和模式形成的基本范式。这个项目的结果将导致有效的数学描述和高效的计算方案的发展，这些问题对材料科学和纳米技术中的小规模物理和化学过程以及生命科学中的定量建模越来越重要。就更广泛的影响而言，该项目通过为自然科学、工程科学和应用数学科学领域的博士后研究人员和博士生提供高级培训，有助于科学劳动力的发展。

Abstract This Focused Research Group brings together researchers from the University of Michigan's Departments of Mathematics, Physics and Chemical Engineering to address important problems of modeling, simulation and analysis for dynamical processes where underlying discreteness plays a non-negligible role in large scale descriptions via deterministic continuum systems (generally systems of ordinary and partial differential equations). This Focused Research Group combines the investigators' expertise in theory, modeling, analysis and scientific computation to study a suite of problems from materials physics, chemical kinetics and the life sciences to elucidate the fundamental scientific issues and develop appropriate quantitative tools to analyze them. The specific problems to be studied are: (1) Mesoscopic mathematical models of wound healing with cell proliferation and migration, and including the biologically important effect of cell-cell adhesion; (2) The application of new and improved simulation techniques, direct solutions of the Becker-Doering equations, and simulation and analysis of stochastic models to investigate the role of microscopic correlations in Ostwald ripening; (3) The development ofanalytic asymptotic methods for accurate reduced descriptions of slow stochastic variables properly incorporating residual fluctuation effects with applications to (bio)chemical reaction networks possessing a wide spectrum of reaction rates; (4) An extension of modeling, analysis and simulation methods developed for simple systems to increasingly complex stochastic models in population biology and epidemiology including epidemics in structured populations and extinction of competing species; (5) Spatial inhomogeneities and reaction-rate variations in the stochastic Fisher-Kolmogorov equation, a fundamental paradigm of front propagation and pattern formation. Results from this project will lead to the development of effective mathematical descriptions and efficient computational schemes for problems of increasing importance for small-scale physical and chemical processes in materials science and nano-technology, and for quantitative modeling in the life sciences. With regard to the even broader impact of this project, it contributes to the development of the scientific workforce by providing advanced training for postdoctoral researchers and doctoral students in the natural, engineering and applied mathematical sciences.