AMC-SS: Equilibrium and Nonequilibrium Statistical Mechanics Applied to Reduced Descriptions of Complex Systems

这个项目的目的是应用统计力学的技术来开发新的方法来模拟具有许多相互作用的变量的复杂动态系统。我们计划设计理论上合理的和计算有效的策略来描述这些系统的一些相关变量的统计平均行为，而不解决其基础的确定性动力学中涉及的所有变量。在这种情况下，一个主要目标是制定统计封闭计划的哈密顿系统与许多自由度。关键的新成分是一个统计封闭计划，选择一个最佳的近似刘维方程的非平衡合奏参数化的几个相关变量。我们的这些想法的实现将强调大偏差分析，系综等效，相变，最可能的状态的非线性稳定性，和计算。推动研究的主要应用是弱湍流系统，在这个意义上，它们在大尺度上表现出相干行为，在小尺度上表现出基本上随机的波动。要考虑的两个主要例子是由广义非线性薛定谔方程控制的色散波湍流和由位涡输运方程控制的地球物理流体流动。brbr该项目解决了当今科学的主要挑战之一，即预测复杂系统的行为。目前复杂性超过我们预测能力的重要物理系统包括分子动力学系统、充分发展的湍流流体和地球的海洋-大气系统。这样的系统的分析和计算的棘手性源于没有基本的动力学模型，但从他们的管理动力系统的高维性加上缺乏信息或数据需要指定这些系统的解决方案。在这些系统的理论和实际处理中需要的是一种有效的方法来预测一些相关宏观变量的行为，而不完全解决所有的动力学变量。在这一奋进中，统计力学提供了占主导地位的范式。 这个研究项目的主要目标之一是通过调整统计力学的方法来解决这些问题，以广泛的一类非平衡系统，这些系统在物理上是适用的，在计算上是易处理的。

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.brbrThis 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.