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

本计划的目的是应用统计力学的技术，开发新的方法来模拟具有许多相互作用变量的复杂动力系统。我们计划设计理论上合理和计算上有效的策略来描述这些系统中一些相关变量的统计平均行为，而不解决其潜在确定性动力学中涉及的所有变量。在这种情况下，主要目标是为具有多自由度的哈密顿系统制定统计闭包格式。关键的新成分是一种统计闭包方案，它在由几个相关变量参数化的非平衡系综中选择Liouville方程的最优逼近。我们对这些思想的实现将强调大偏差分析、集合等效、相变、最可能状态的非线性稳定性和计算。推动这项研究的主要应用是弱湍流系统，因为它们在大尺度上表现出连贯的行为，在小尺度上表现出本质上的随机波动。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。这个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。在复杂性目前超过我们预测能力的重要物理系统中，有分子动力系统、完全发展的湍流流体和地球的海洋-大气系统。这类系统难以分析和计算的原因不是因为缺乏潜在的动力模型，而是因为它们的控制动力系统的高维性，加上缺乏指定这些系统的解决方案所需的信息或数据。对这些系统的理论和实际处理需要的是一种有效的方法来预测几个相关的宏观变量的行为，而不完全解决所有的动态变量。在这一努力中，统计力学提供了主导范式。本研究项目的主要目标之一是通过将统计力学方法应用于广泛的非平衡系统来解决这些问题，这些系统在物理上是适用的，在计算上是可处理的。

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.