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

0202309Ellis这个项目的主要关注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。这项研究将考虑一系列问题，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不等价性；(2)研究在各种系综的大偏差分析中产生的随机过程的中心极限型涨落；(3)基于高斯测量的大偏差结果，研究由非线性薛定谔方程控制的孤子湍流的渐近分析的新方法；(4)基于系综不等价性研究中的思想，对Arnold关于一类控制湍流现象的偏微分方程流的非线性稳定性的经典结果进行了改进；(5)随机游走模型的分析，该模型应用于排队论、通信理论和随机逼近。在问题(1)-(4)的背景下，首席研究员将开发可应用于物理科学中尚未解决的重大问题之一的数学工具，即对湍流流动的理解。在这个项目中，通过基于统计力学形式的复杂的概率模型来研究湍流固有的不可预测性和随机性。使用大偏差理论和概率论、凸分析、偏微分方程组等领域的技术对模型进行分析。

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.