量子输运的多尺度建模和计算

Quantum transport is very important in various fields of scientific research and engineering, especially in recent years with the discovery of graphene, topological insulator and the experiment observation of the quantum anomalous Hall effect, it becomes a new research hotspot...This project aims to build multi-scale models and develop efficient numerical methods for quantum transport. We want to derive multi-scale model of quantum transport in new materials such as graphene, topological insulators, and give the theoretical analysis of the models. For the multi-scale transport models which have the diffusion limit, we want to use asymptotic preserving scheme to solve them, and prove the stability of the scheme. The coupling of the semi-classical limit and small scale model are used to design fast and effective numerical schemes for more general models. The de Broglie-Bohm theory is used to build the quantum kinetic transport model with Bohm potential for many-body system, and the high order numerical methods are constructed to solve these models. The asymptotic method such as the Gauss beam method are used to solve the surface hopping problem at the Dirac point. The key point is how to design the interface conditions at of these asymptotic methods at band intersection point and estimate the asymptotic convergence rate of these methods...We want to construct the multi-scale models which can capture quantum effect of quantum transport, and the computational cost are of the same order to the classical mechanics.

量子输运在科学研究与工程实践的各个领域有着广泛的应用，尤其是近年来随着石墨烯、拓扑绝缘体以及量子反常霍尔效应的发现，成为了新的研究热点。..本项目拟建立描述新兴材料如石墨烯、拓扑绝缘体等量子输运的多尺度模型，并对模型进行理论分析。对具有扩散极限的多尺度输运模型，设计渐进保持的格式，并给出算法的稳定性的证明。对更一般的模型，则通过半经典极限和小尺度模型耦合的思想来设计快速有效的数值格式。对多粒子体系的量子输运，尝试de Broglie-Bohm理论来简化模型，得到带有Bohm势的量子输运模型，并给出高阶的计算方法。用高斯束方法来求解Dirac点量子输运问题，导出高斯束在能带相交点的界面条件，并给出高斯束方法的渐进收敛速度的理论估计。..本项目想得到既能有效捕捉量子输运的量子效应，而计算量和经典力学基本相当的多尺度模型和算法。

量子输运在众多的科学研究领域有着广泛的应用。量子输运现象具有量子效应不能用经典力学来解释，且不依赖于纳米器件的尺度。..本项目的研究内容是量子输运问题的建模和计算，想建立能够有效的描述新兴材料如拓扑绝缘体、石墨烯等的量子输运现象的多尺度模型，并对这些模型设计有效的算法，使得既能捕捉到我们关心的量子效应，且其计算量和经典力学的计算量基本相当。..经过课题组四年的研究工作，得到了Gaussian beam和Schrodinger方程的耦合模型，并用于石墨烯的数值模拟。对量子Boltzmann方程证明了其在低温状态下解长时间收敛到 Bose-Einstein分布。用量身定做的有限点方法求解跟量子输运相关的奇异摄动问题，得到了一致收敛的算法，使得既能捕捉到量子效应，且其计算量和经典力学的计算量基本相当。对多尺度电磁场散射问题进行了深入研究，设计了快速有效的算法。设计了求解时谐的多尺度弹性波方程的间断有限元方法。..课题组共发表论文10篇，其中期刊论文8篇，SCI收录6篇，会议论文2篇。基本完成了项目的目标。