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

0202309 Ellis该项目的主要重点是湍流统计力学模型的渐近分析，使用大偏差理论以及其他分析领域的强大技术。研究内容包括：（1）详细研究各种湍流模型的统计力学系综的等价性和非等价性，（2）研究各种系综的大偏差分析中随机过程的中心极限型涨落，（3）研究各种系综的大偏差分析中随机过程的中心极限型涨落，（4）研究各种系综的大偏差分析中随机过程的中心极限型涨落。（3）基于高斯测度的大偏差结果，对非线性Schroedinger方程控制的孤子湍流的渐近分析提出了一种新的方法;（4）一种改进，基于在研究系综的不等价性时产生的思想，Arnold关于一类控制湍流现象的偏微分方程的非线性稳定性的经典结果;（5）对随机游走模型的分析，其应用于传播理论、通信理论和随机逼近。 在问题（1）-（4）的背景下，主要研究者将开发数学工具，可以应用于物理科学的一个伟大的未解决的问题，这是湍流的理解。在这个项目中，固有的不可预测性和随机性的湍流研究通过复杂的概率模型的基础上的形式主义的统计力学。使用大偏差理论和概率论、凸分析、偏微分方程和其他领域的技术来分析模型。

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.