Computation of the Semiclassical Limit of Schroedinger's Equation, Anisotropic Grain Growth, and Epitaxial Growth Using Kinetic Monte Carlo

该提案涉及三个项目，其中每个项目都是由先前的NSF支持构建的。第一个项目使用我们最近开发的多相变分水平集框架模拟二维和三维的晶界运动。计划扩展这个公式，以允许晶粒之间任意表面张力的可能性。这将通过将这个公式与最小化动作相结合来实现。第二个项目是关于薛定谔方程半经典极限的有效计算。提出的算法是基于这样的观察：如果用高斯波包的变换变换薛定谔方程，就会得到另一个更适合在半经典极限方案中计算的薛定谔方程。第三个项目涉及利用动力学蒙特卡罗（KMC）建模和有效模拟外延生长。我们计划模拟液滴外延和异质外延生长。在每种情况下，我们的目标是开发高效的KMC算法，以允许在以前不可能的时间和长度上进行模拟，从而促进模型开发并允许与实验进行比较。每一个拟议的项目都有可能对具有根本性和技术重要性的问题产生重大影响。外延生长在纳米尺度和中尺度上都有影响，因此在科学上很有趣。这在技术上是相关的，因为量子点材料是用这种方式制造的。我们提出的技术将大大提高仿真速度，从而促进模型的开发。利用曲率流动研究晶界运动是应用数学和计算数学中的一个经典问题，在材料科学中具有重要意义，因为多晶物质的各种特性在很大程度上取决于晶粒图案的细节。对薛定谔方程半经典极限的快速模拟可以为化学反应动力学、分子表面散射和光解等提供更深入的了解。

This proposal involves three projects where each one builds from prior NSF support. This first project involves the simulation of grain boundary motion in two and three dimensions using our recently developed multiphase variational level set framework. It is planned to extend this formulation to allow for the possibility of arbitrary surface tension between grains. This will be accomplished by combining this formulation with minimizing movements. The second project concerns the efficient computation the semi-classical limit of the Schroedinger equation. The proposed algorithm is based on the observation that if one transforms the Schroedinger equation using a transformation inspired by Gaussian wave packets, one arrives at another Schroedinger equation that is much more amenable to computation in the semiclassical limit project. The third project concerns modeling and efficient simulation of epitaxial growth using kinetic Monte Carlo (KMC). We plan to model and simulate both liquid drop epitaxy and heteroepitaxial growth. In each of these cases we aim to develop highly efficient KMC algorithms to allow simulation on time and lengths not previously possible, thereby facilitating model development and allowing comparison with experiments.Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Epitaxial growth is scientifically interesting since it has effects on both nanoscales and mesoscales. It is technologically relevant since quantum dot materials are made in this way. Our proposed techniques will greatly increase the simulation speed thereby facilitating model development. The study of grain boundary motion using curvature flow is a classic problem in applied and computational mathematics which has importance in material science since various properties of polycrystalline substances can crucially depend on the details of the grain patterns. The fast simulation of the semi-classical limit of the Schroedinger equation could provide deeper insight into chemical reaction dynamics, molecular-surface scattering, and photodissociation, for example.