Research in Large Deviations and Applications to Statistical Mechanical Models of Turbulence

大偏差研究及其在湍流统计力学模型中的应用

• 批准号: 0202309
• 负责人: Richard Ellis
• 金额: $ 22.77万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202309&HistoricalAwards=false
• 关键词: Research; Large; Deviations; Applications; Statistical

0202309Ellis The main focus of this project is on the asymptotic analysis of statistical mechanical models of turbulence, using powerful techniques in the theory of large deviations as well as other areas of analysis. The research will consider a number of problems, including the following: (1) a detailed study of the equivalence and nonequivalence of statistical mechanical ensembles for a wide range of models of turbulence; (2) an investigation of central-limit-type fluctuations of the stochastic processes arising in the large deviation analysis of the various ensembles; (3) a new approach, based on large deviation results for Gaussian measures, to the asymptotic analysis of soliton turbulence governed by the nonlinear Schroedinger equation; (4) a refinement, based on ideas arising in the study of nonequivalence of ensembles, of classical results due to Arnold on the nonlinear stability of flows for a class of partial differential equations governing turbulence phenomena; (5) an analysis of a random walk model having applications to queueing theory, communication theory, and stochastic approximation. In the context of problems (1)-(4) the principal investigator will develop mathematical tools that can be applied to one of the great unsolved problems of the physical sciences, which is the understanding of turbulent fluid flows. In this project the inherent unpredictability and randomness of turbulent flows are studied via sophisticated probabilistic models based on the formalism of statistical mechanics. The models are analyzed using techniques in the theory of large deviations and probability theory, convex analysis, partial differential equations, and other areas.

0202309 ellis该项目的主要重点是使用强大的大偏差理论以及其他分析领域的强大技术对湍流的统计机械模型的渐近分析。该研究将考虑许多问题，包括以下问题：（1）对广泛的湍流模型的统计机械集合的等效性和无数的详细研究； （2）在对各种合奏的大偏差分析中产生的随机过程的中央限制型波动的研究； （3）一种基于高斯措施的较大偏差结果的新方法，对非线性schroedinger方程控制的独感湍流的渐近分析； （4）基于在合奏的不重要性研究中产生的思想，这是由于阿诺德（Arnold）对一类偏微分方程的非线性稳定性而导致的经典结果，该方程涉及湍流现象； （5）对具有排队理论，交流理论和随机近似的随机步行模型的分析。 在问题的背景下（1） - （4）主要研究者将开发数学工具，这些工具可以应用于物理科学的一个巨大的未解决问题之一，这是对湍流流动的理解。在该项目中，基于统计力学的形式主义，通过复杂的概率模型来研究湍流的固有不可预测性和随机性。使用大偏差和概率理论，凸分析，部分微分方程和其他领域的技术中的技术进行分析。