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

摘要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，以解决动态过程的建模、模拟和分析等重要问题，其中潜在的离散性在通过确定性连续系统(通常是常微分方程组和偏微分方程组)的大规模描述中起着不可忽视的作用。这个重点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，研究了一系列来自材料物理、化学动力学和生命科学的问题，以阐明基本的科学问题，并开发适当的量化工具进行分析。要研究的具体问题是：(1)创伤愈合与细胞增殖和迁移的介观数学模型，并包括细胞-细胞黏附的生物学重要影响；(2)新的和改进的模拟技术的应用，Becker-Doering方程的直接解，以及随机模型的模拟和分析，以研究微观相关性在Ostwald成熟中的作用；(3)发展解析渐近方法，准确地简化描述慢速随机变量，适当地结合剩余涨落效应，并应用于具有广泛反应速率的(生物)化学反应网络；(4)将为简单系统开发的建模、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方程的空间不均匀性和反应速率变化，该方程是前沿传播和模式形成的基本范式。该项目的结果将导致为材料科学和纳米技术中日益重要的小规模物理和化学过程以及生命科学中的定量建模问题开发有效的数学描述和高效的计算方案。关于该项目的更广泛影响，它通过为博士后研究人员和博士后提供自然科学、工程学和应用数学科学方面的高级培训，促进了科学队伍的发展。

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.