Phase Transitions in Smooth Dynamical Systems

平滑动力系统中的相变

• 批准号: 0400687
• 负责人: Omri Sarig
• 金额: $ 12万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400687&HistoricalAwards=false
• 关键词: Phase; Transitions; Smooth; Dynamical; Systems

Sarig conduct a project of research, education and outreach in ergodic theory and dynamical systems, analyzing the stochastic nature of non-compact or non-uniformly hyperbolic systems. Building on previous work of the PI and others, the proposal identifies an analogy between certain phenomena associated with such systems and critical phenomena in statistical physics (`phase transitions'). The PI proposes a detailed program for utilizing this analogy to explore the ergodic theoretic effects of non-uniform hyperbolicity or non-compactness systematically. The intellectual merit of the project is that it uses an original approach to study an important class of dynamical systems: Most of the dynamical systems in applications are not uniformly hyperbolic, or are hyperbolic only on a non-compact part of their phase space. The broader impact is a component aimed at three groups: undergraduate students, graduate students, and experts. The mathematics department in the Pennsylvania State University has an existing program aimed at injecting research into the learning process at an early stage: the Mathematics Advanced Studies Semesters (MASS) program. The PI intends to develop a topic course to be taught within this program, with the purpose of attracting undergraduates to join a graduate program in mathematics. The graduate component consists of training Ph.D. students, which the PI will endeavor to attract. The expert component is a plan to write a research monograph, concerning the recent results of the PI on topics related to the proposal. The title of the suggested monograph is `The thermodynamic formalism of countable Markov shifts'.

Sarig进行遍历理论和动力系统的研究，教育和推广项目，分析非紧或非均匀双曲系统的随机性质。在PI和其他人以前工作的基础上，该提案确定了与此类系统相关的某些现象与统计物理学中的临界现象（“相变”）之间的类比。PI提出了一个详细的计划，利用这个类比系统地探索遍历理论的非一致双曲性或非紧性的影响。 该项目的智力价值在于它使用了一种新颖的方法来研究一类重要的动力系统：大多数应用中的动力系统不是一致双曲的，或者只在相空间的非紧部分是双曲的。 更广泛的影响是针对三个群体的组成部分：本科生，研究生和专家。宾夕法尼亚州立大学的数学系有一个现有的项目，旨在在早期阶段将研究注入学习过程：数学高级研究学期（MASS）项目。PI打算开发一个主题课程，在这个程序中教授，吸引本科生加入数学研究生课程的目的。研究生部分包括培养博士。学生，这是PI将奋进吸引。专家部分是一项撰写研究专著的计划，涉及PI在与提案相关的主题上的最新结果。建议的专着的标题是“可数马尔可夫位移的热力学形式主义”。