Mathematical Sciences: Scaling Limits of Nongradient Dynamics

关于随机动力学的第一个问题是松弛到平衡的速率。人们可以粗略地将这些速率分为两类：指数松弛和幂律松弛。本研究的主要目的是研究具有幂律衰减的动力学。第一个目标是严格推导宏观尺度极限方程，然后研究对这些方程的各种修正。在过去的几年里，已经开发了几种方法来解决这些问题。这些方法的主要限制之一是称为梯度条件的技术条件。研究者将研究几种非梯度模型，特别是在有限温度下晶格气的流体力学极限和弛豫到平衡的速率。我们也将考虑具有无规场的格子气的一个相关问题。最后，研究人员将研究与时间尺度比流体动力学尺度长有关的问题。这些是特别重要的，因为不可压缩的Navier-Stokes方程被认为是哈密顿系统的长时间（扩散）标度极限。 动力系统的一个基本问题是确定系统接近平衡点的速率。在这项研究中，研究人员将研究动态缓慢放松到平衡，大多数自然过程遵循。主要目标是确定这些过程平衡的精确弛豫速率，并导出控制弛豫宏观行为的宏观方程，然后研究对这些方程的各种修正。在过去的几年中，已经开发了几种方法来解决这些问题的几种类型的模型。但是这些方法的一些主要限制阻碍了对更真实模型的研究。研究人员将研究几种模型，以扩展现有技术并解决一些关键技术难题。感兴趣的模型包括：有限温度下的晶格气体，晶格气体中随机杂质的影响，以及模拟不可压缩Navier-Stokes方程的离散速度模型。

The very first question concerning stochastic dynamics is the rate of relaxation to the equilibrium. One can roughly divide these rates into two categories: the exponential relaxation and the power law relaxation. The main objective of this research is to study dynamics with the power law decay. The first goal is to derive the macroscopic scaling limiting equations rigorously and then to study various corrections to these equations. In the past few years, several methods have been developed for these problems. One of the major restrictions to these methods is a technical condition called the gradient condition. The investigator will study several nongradient models, in particular, the hydrodynamical limit and the rate of relaxation to equilibrium of lattice gas at finite temperature. A related problem for lattice gas with random field will also be considered. Finally, the investigator will study problems related to a time scale longer than the hydrodynamical scale. These are particularly important because the incompressible Navier-Stokes equation is believed to be the long time (diffusive) scaling limit of Hamiltonian systems. A basic problem concerning dynamical system is to determine the rate of approach to the equilibrium. In this research the investigator will study dynamics with slow relaxation to equilibrium, which most natural processes follow. Major goals are to determine the precise rate of relaxation to the equilibrium of these processes and to derive macroscopic equations governing the macroscopic behavior of relaxations and then to study various corrections to these equations. In the past few years, several methods have been developed for these problems for several types of models. But some major restrictions of these methods prohibit the study of more realistic models. The investigator will study several models to extend the current technology and to address some key technical difficulties. Models of interest include: the lattice gases at finite temperature, the effect of random impurities in a lattice gas, and the discrete velocity model which simulates the incompressible Navier-Stokes equation.