Lee Tsung-Yen, Lu Chun-Yaung, Chou Chia-Chun
Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan.
Texas Advanced Computing Center, The University of Texas at Austin, Austin, Texas 78758, United States.
J Phys Chem A. 2021 Jan 14;125(1):476-491. doi: 10.1021/acs.jpca.0c09525. Epub 2020 Dec 29.
The moving boundary truncated grid (TG) method, previously developed to integrate the time-dependent Schrödinger equation and the imaginary time Schrödinger equation, is extended to the time evolution of distribution functions in phase space. A variable number of phase space grid points in the Eulerian representation are used to integrate the equation of motion for the distribution function, and the boundaries of the TG are adaptively determined as the distribution function evolves in time. Appropriate grid points are activated and deactivated for propagation of the distribution function, and no advance information concerning the dynamics in phase space is required. The TG method is used to integrate the equations of motion for phase space distribution functions, including the Klein-Kramers, Wigner-Moyal, and modified Caldeira-Leggett equations. Even though the initial distribution function is nonnegative, the solutions to the Wigner-Moyal and modified Caldeira-Leggett equations may develop negative basins in phase space originating from interference effects. Trajectory-based methods for propagation of the distribution function do not permit the formation of negative regions. However, the TG method can correctly capture the negative basins. Comparisons between the computational results obtained from the full grid and TG calculations demonstrate that the TG method not only significantly reduces the computational effort but also permits accurate propagation of various distribution functions in phase space.
先前为积分含时薛定谔方程和虚时薛定谔方程而开发的移动边界截断网格（TG）方法，被扩展到相空间中分布函数的时间演化。在欧拉表示中，使用可变数量的相空间网格点来积分分布函数的运动方程，并且随着分布函数随时间演化，TG的边界被自适应地确定。为了分布函数的传播，适当的网格点被激活和停用，并且不需要关于相空间中动力学的预先信息。TG方法用于积分相空间分布函数的运动方程，包括克莱因 - 克莱默斯方程、维格纳 - 莫亚尔方程和修正的卡尔德雷拉 - 莱格特方程。尽管初始分布函数是非负的，但维格纳 - 莫亚尔方程和修正的卡尔德雷拉 - 莱格特方程的解可能会在相空间中由于干涉效应而产生负区域。基于轨迹的分布函数传播方法不允许形成负区域。然而，TG方法可以正确地捕捉到负区域。从全网格计算和TG计算获得的计算结果之间的比较表明，TG方法不仅显著减少了计算量，而且还允许在相空间中精确传播各种分布函数。
