Mathematical Sciences: Statistical Mechanics of Spatially Extended Dynamical Systems

9501045 Kupiainen该项目使用以前在统计力学中开发的方法来研究具有空间扩展吸引子的动力系统。第一组问题涉及非线性抛物型偏微分方程中图案和阵面的形成，重整化群将被用来研究Cahn-Hilliard方程和相关方程中阵面和图案的普遍性质。第二组问题涉及混沌无限维动力学。用展开方法研究耦合映象格子和概率元胞自动机中的时空混沌和相变现象。扩展的动力系统模拟了宏观系统中存在的规则和混沌现象，宏观系统由大量相同的部分组成，如液体、气体和一些生物和生态系统。将研究规则模式(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)的形成，以及在时间和空间上表现为不规则行为的不规则、混乱现象(例如，流体的湍流运动)。以前为数学物理其他领域中的问题开发的方法将被用来研究这些现象的普遍方面，即对所使用的特定模型相当不敏感的方面。***

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. ***