The purpose of this project is to apply the techniques of statistical mechanics to develop new approaches to modeling complex dynamical systems with many interacting variables. We plan to devise theoretically justified and computationally effective strategies for describing the statistical average behavior of a few relevant variables for such systems without resolving all of the variables involved in their underlying deterministic dynamics. In this general context a principal objective is to formulate statistical closure schemes for Hamiltonian systems with many degrees of freedom. The key new ingredient is a statistical closure scheme that selects an optimal approximation to the Liouville equation among nonequilibrium ensembles parameterized by a few relevant variables. Our implementation of these ideas will emphasize large deviation analysis, ensemble equivalence, phase transitions, nonlinear stability of most probable states, and computations. The main applications that drive the research are systems that are weakly turbulent in the sense that they exhibit coherent behavior on large scales and essentially random fluctuations on small scales. The two main examples to be considered are dispersive wave turbulence governed by generalized nonlinear Schroedinger equations and geophysical fluid flows governed by potential vorticity transport equations.
This project addresses one of the major challenges in science today, which is to predict the behavior of complex systems. Among the important physical systems whose complexity currently exceeds our predictive capacity are molecular dynamical systems, fully-developed turbulent fluids, and the Earth's ocean-atmosphere system. The intractability of such systems to analysis and computation stems not from the absence of underlying dynamical models, but from the high dimensionality of their governing dynamical systems coupled with lack of information or data needed to specify the solutions of those systems. What is needed in theoretical and practical treatments of these systems is an effective way to predict the behavior of a few relevant macroscopic variables without completely resolving all the dynamical variables. In this endeavor, statistical mechanics supplies the dominant paradigms. One of the main goals of this research project is to address these issues by adapting the methods of statistical mechanics to a wide class of nonequilibrium systems that are both physically applicable and computationally tractable.
