Mathematical Sciences: Statistical Mechanics of Spatially Extended Dynamical Systems

数学科学：空间扩展动力系统的统计力学

• 批准号: 9501045
• 负责人: Antii Kupiainen
• 金额: $ 7.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501045&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Statistical; Mechanics; Spatially

9501045 Kupiainen The project uses methods previously developed in Statistical Mechanics for the study of dynamical systems with spatially extended attractors. The first set of problems deals with the formation of patterns and fronts in nonlinear parabolic PDE's. The Renormalization Group will be used to study universal properties of fronts and patterns in the Cahn-Hilliard and related equations. The second set of problems deals with chaotic infinite dimensional dynamics. Expansion methods will be used to study the phenomena of spatio-temporal chaos and phase transitions in coupled map lattices and probabilistic cellular automata. Extended dynamical systems model regular and chaotic phenomena that are present in macroscopic systems that are composed of a very large number of identical parts such as liquids, gases and some biological and ecological systems. Both the formation of regular patterns (real life examples are waves in oceans, convective phenomena in atmosphere and patterns in clouds) and irregular, chaotic phenomena which exhibit chaos as irregular behaviour both in time and in space (for example turbulent motion of fluids) will be studied. Previously developed methods for problems in other areas of mathematical physics will be used to study universal aspectsof these phenomena, i.e. aspects that are rather insensitive to specific models that are being used. ***

这个项目使用以前在统计力学中发展起来的方法来研究具有空间扩展吸引子的动力系统。第一组问题处理非线性抛物型偏微分方程中图形和锋面的形成。重整化群将用于研究卡恩-希利亚德方程和相关方程中锋面和模式的普遍性质。第二组问题处理混沌无限维动力学问题。展开方法将用于研究耦合映射格和概率元胞自动机中的时空混沌和相变现象。扩展动力系统对存在于由大量相同部分组成的宏观系统（如液体、气体和某些生物和生态系统）中的规则和混沌现象进行建模。规则模式的形成（现实生活中的例子是海洋中的波浪，大气中的对流现象和云中的模式）和不规则的混沌现象，这些现象在时间和空间上都表现出不规则的混沌行为（例如流体的湍流运动）。以前开发的数学物理其他领域问题的方法将用于研究这些现象的普遍方面，即对正在使用的特定模型相当不敏感的方面。＊＊＊