Scaling Limits for Stochastic and Quantum Dynamics

随机和量子动力学的标度极限

• 批准号: 9703752
• 负责人: Horng-Tzer Yau
• 金额: $ 28.2万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2001-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703752&HistoricalAwards=false
• 关键词: Scaling; Limits; Stochastic; Quantum; Dynamics

9703752 Yau The incompressible Navier-Stokes equation have been proved as the hydrodynamical limit equation of stochastic lattice gas models in dimension 3. The first project investigates the following three related topics: (i) the scaling limit of tagged particles, (ii) the equilibrium fluctuation and (iii) the appropriate time scale for dimensions 1 and 2. The first problem emphasizes the individual behavior of a typical particle instead of the collective behavior of all the particles. The second problem is a question about the central limit theorem. The third problem addresses the time scale in dimensions 1 and 2, which is conjectured to be very different from the diffusive scale in dimension 3. The second project studies the relaxation rates of the Kawasaki dynamics of Gibbs states in infinite volume. This is believed to be a power law. The method is based on the Poincare inequalities (spectral gap), logarithmic Sobolev inequalities and some entropy estimates. The third project concerns the scaling limit of a quantum particle in a random potential. There are two limiting cases: the low density limit and the weak coupling limit. In both cases, the phase space density of the quantum evolution defined through the Wigner transform or the coherent state is expected to converge weakly to a linear Boltzmann equation with collision kernel given by the quantum scattering cross section. This research addresses in particular the following two questions: (1) What is the typical behavior of an individual particle in a fluid? Though the collective behavior of fluid has been studied intensively, the important question of the propagation of individual particles in the fluid has not. The project will study this question with stochastic models which are believed to capture the essential behavior of the fluid. (2) How does a wave travel in random media. It is believed that with high disorder, a wave in random media becomes loca lized. The important question is to analyze the low disorder region when a wave propagates. This can be considered as a model for conduction of current in a semiconductor, or propagation of radio waves or seismic waves. The practical equation governing all these diverse phenomena is the Boltzmann equation. This study will try to validate the Boltzmann equation from more basic models involving wave equations in random media and to understand its next order corrections or fluctuations.

小行星9703752 本文证明了不可压Navier-Stokes方程为 三维随机格子气模型流体动力学极限方程第一 项目调查了以下三个相关主题：（一）标记的缩放限制 粒子，（ii）平衡涨落和（iii）维度的适当时间尺度 1和2. 第一个问题强调的是一个典型粒子的个体行为 所有粒子的集体行为。第二个问题是关于中心极限定理的问题。第三个问题涉及1维和2维的时间尺度， 其被证明与3维中的扩散尺度非常不同。的 第二个项目研究吉布斯态的川崎动力学的弛豫速率， 无穷大的体积这被认为是幂律。该方法是基于庞加莱 不等式（谱隙），对数Sobolev不等式和一些熵估计。 第三个项目涉及量子粒子在随机势中的标度极限。 有两种极限情况：低密度极限和弱耦合极限。无论是 在某些情况下，通过Wigner定义的量子演化的相空间密度 变换或相干态预期弱收敛到线性玻尔兹曼 碰撞核由量子给出的方程 散射截面 本研究主要探讨了以下两个问题：（1）什么是 流体中单个粒子的典型行为尽管流体的集体行为 已经被深入研究，个体繁殖的重要问题 流体中的粒子没有。该项目将使用随机模型研究这个问题 其被认为捕获流体的基本行为。 (2)一个波浪 在随机介质中传播。人们相信，在高度无序的情况下，随机介质中的波 变得本地化。重要的问题是分析低无序区时， 波传播。这可以被认为是一个模型的传导电流在一个 半导体或无线电波或地震波的传播。实用方程式 控制所有这些不同现象的是玻尔兹曼方程。这项研究将试图 从涉及波动方程的更基本模型验证玻尔兹曼方程， 随机介质，并了解其下一阶校正或波动。