Asymptotic Dynamics for Stochastic and Quantum Dynamics

[307295]量子粒子在随机介质中的动力学是由随机薛定谔方程控制的。在动力学极限下，该方程解的相空间分布收敛于线性玻尔兹曼方程。在动力学尺度之外，人们期望玻尔兹曼方程在三维或更高的维度上仍然是正确的，只要扩散系数被重新归一化。第一个项目的目的是在比动力学尺度更长的时间尺度上研究这种重整化。第二个项目涉及晶格气体模型的粘度。最近，PI建立了二维非对称简单不相容过程的扩散系数的发散率。本项目的目标是将这一结果推广到一维情况和以Navier-Stokes方程为形式极限的晶格气体模型。关于半导体中电子导电性的基本问题是导电是如何发生的。随着技术发展到半经典理论不再适用的阶段，量子效应将占据主导地位，一个数学严谨的理论至关重要。第一个项目旨在建立基于玻尔兹曼方程的标准半经典理论的量子修正。这将首次提供对这些量子动力学效应的严格理解。第二个项目研究了晶格气体模型对流体的有效性。这些模型不仅广泛用于模拟Navier-Stokes方程，而且是非平衡统计物理的重要模型。这里的目标是为这些模型建立粘度的发散率。这将提供关于纳维-斯托克斯方程如何很好地模拟二维流体的第一个线索。

0307295Yau The dynamics of a quantum particle in a random media is governed by a random Schrodinger equation. In the kinetic limit, the phase space distribution of a solution to this equation converges to a linear Boltzmann equation. Beyond the kinetic scale, one expects that the Boltzmann equation is still correct in dimension three or higher provided that the diffusivity is renormalized. The first project aims to study this renormalization for time scales longer than the kinetic scale. The second project concerns the viscosity of the lattice gas models. Recently the PI has established the divergence rate for the diffusion coefficient for the asymmetric simple exclusion process in dimension two. The goal of this project is to extend this result to the dimension one case and to the lattice gas models with the Navier-Stokes equation as the formal limit. The fundamental question regarding the conductivity of electrons in semiconductors is how conduction occurs. As the technology is reaching the stage that the semi-classical theory no longer applies, the quantum effect will dominate and a mathematical rigorous theory is of fundamental importance. The first project aims to establish the quantum corrections of the standard semi-classical theory based on the Boltzmann equation. This will provide the first rigorous understanding of these quantum dynamical effects. The second project investigates the validity of the lattice gas models for fluids. These models, besides being widely used to simulate the Navier-Stokes equations, are important models for non-equilibrium statistical physics. The goal here is to establish the divergence rate of the viscosity for these models. This will give the first clue concerning how well the Navier-Stokes equation models the two-dimensional fluid.