Scaling Limits for Stochastic and Quantum Dynamics

[9703752]证明了不可压缩的Navier-Stokes方程是三维随机点阵气体模型的水动力极限方程。第一个项目研究以下三个相关主题：(i)标记粒子的缩放极限，（ii）平衡波动和（iii）维度1和2的适当时间尺度。第一个问题强调一个典型粒子的个体行为，而不是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。第三个问题处理维度1和2的时间尺度，这被推测与维度3的扩散尺度有很大不同。第二个项目研究了无限体积下吉布斯态川崎动力学的弛豫速率。这被认为是幂律。该方法基于庞加莱不等式（谱间隙）、对数Sobolev不等式和一些熵估计。第三个项目涉及随机势中的量子粒子的缩放极限。有两种极限情况：低密度极限和弱耦合极限。在这两种情况下，通过维格纳变换或相干态定义的量子演化相空间密度都有望弱收敛于由量子散射截面给出的具有碰撞核的线性玻尔兹曼方程。本研究主要解决以下两个问题：(1)流体中单个粒子的典型行为是什么？虽然流体的集体行为已被深入研究，但单个粒子在流体中传播的重要问题尚未得到深入研究。该项目将用随机模型来研究这个问题，这些模型被认为能够捕捉流体的基本行为。(2)波如何在随机介质中传播。人们认为，在高度无序的情况下，波在随机介质中变得局域化。重要的问题是分析波传播时的低无序区。这可以看作是半导体中电流传导的模型，或者是无线电波或地震波的传播模型。控制所有这些不同现象的实际方程是玻尔兹曼方程。本研究将尝试从涉及随机介质中的波动方程的更基本模型中验证玻尔兹曼方程，并了解其下一阶修正或波动。

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.