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

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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.