Absorbing Boundary Conditions for Time-dependent Schr\"{o}dinger equations: A Density-matrix Formulation

This paper presents some absorbing boundary conditions (ABC) for simulations based on the time-dependent density-functional theory (TDDFT). The boundary conditions are expressed in terms of the elements of the density-matrix, and it is derived from the full model over a much larger domain. To make the implementation much more efficient, several approximations for the convolution integral will be constructed with guaranteed stability. These approximations lead to modified density-matrix equations at the boundary. The effectiveness is examined via numerical tests

Markovian Embedding Procedures for Non-Markovian Stochastic Schr\"{o}dinger Equations

We present embedding procedures for the non-Markovian stochastic Schr\"{o}dinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schr\"{o}dinger equations. The stochastic Schr\"{o}dinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and we derive the corresponding generalized quantum master equation for the density-matrix. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schr\"{o}dinger equations. The asymptotic behavior of the spectral density in the low frequency regime is preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example

Coarse-graining molecular dynamics models using an extended Galerkin projection

We present a new framework for coarse-graining molecular dynamics models for crystalline solids. The reduction method is based on a Galerkin projection to a subspace, whose dimension is much smaller than that of the full atomistic model. The subspace is expanded by adding more coarse-grain variables near the interface between lattice defects and the surrounding regions. This effectively minimizes reflection of phonons at the interface. In this approach, there is no need to pre-compute the memory function in the generalized Langevin equations, a typical model of interface conditions. Moreover, the variational formulation preserves the stability of mechanical equilibria

On the Asymptotic Behavior of the Kernel Function in the Generalized Langevin Equation: A One-dimensional lattice model

We present some estimates for the memory kernel function in the generalized Langevin equation, derived using the Mori-Zwanzig formalism from a one-dimensional lattice model, in which the particles interactions are through nearest and second nearest neighbors. The kernel function can be explicitly expressed in a matrix form. The analysis focuses on the decay properties, both spatially and temporally, revealing a power-law behavior in both cases. The dependence on the level of coarse-graining is also studied

Stable absorbing boundary conditions for molecular dynamics in general domains

A new type of absorbing boundary conditions for molecular dynamics simulations are presented. The exact boundary conditions for crystalline solids with harmonic approximation are expressed as a dynamic Dirichlet- to-Neumann (DtN) map. It connects the displacement of the atoms at the boundary to the traction on these atoms. The DtN map is valid for a domain with general geometry. To avoid evaluating the time convo- lution of the dynamic DtN map, we approximate the associated kernel function by rational functions in the Laplace domain. The parameters in the approximations are determined by interpolations. The explicit forms of the zeroth, first, and second order approximations will be presented. The stability of the molecular dynamics model, supplemented with these absorbing boundary conditions is established. Two numerical simulations are performed to demonstrate the effectiveness of the methods.Comment: 25 pages, 4 figure

A study on the quasiconinuum approximations of a one-dimensional fracture model

We study three quasicontinuum approximations of a lattice model for crack propagation. The influence of the approximation on the bifurcation patterns is investigated. The estimate of the modeling error is applicable to near and beyond bifurcation points, which enables us to evaluate the approximation over a finite range of loading and multiple mechanical equilibria

The Mori-Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics

Energy transport equations are derived directly from full molecular dynamics models as coarse-grained description. With the local energy chosen as the coarse-grained variables, we apply the Mori-Zwanzig formalism to derive a reduced model, in the form of a generalized Langevin equation. A Markovian embedding technique is then introduced to eliminate the history dependence. In sharp contrast to conventional energy transport models, this derivation yields {\it stochastic} dynamics models for the spatially averaged energy. We discuss the approximation of the random force using both additive and multiplicative noises, to ensure the correct statistics of the solution

The strong convergence of operator-splitting methods for the Langevin dynamics model

We study the strong convergence of some operator-splitting methods for the Langevin dynamics model with additive noise. It will be shown that a direct splitting of deterministic and random terms, including the symmetric splitting methods, only offers strong convergence of order 1. To improve the order of strong convergence, a new class of operator-splitting methods based on Kunita's solution representation are proposed. We present stochastic algorithms with strong orders up to 3. Both mathematical analysis and numerical evidence are provided to verify the desired order of accuracy

On Consistent Definitions of Momentum and Energy Fluxes for Molecular Dynamics Models with Multi-body Interatomic Potentials

In this paper, we propose a two-level criteria to check the consistency of the definitions of continuum quantities in Molecular Dynamics. As examples, we follow the control- volume approach, derive the definitions of the tractions and energy fluxes for EAM potential and Tersoff potential, and provide the pseudo code the computing. Then, we verify the consistency of the definitions by analytical and numerical methods.Comment: 27 pages, 12 figure

Traction Boundary Conditions for Molecular Static Simulations

This paper presents a consistent approach to prescribe traction boundary conditions in atomistic models. Due to the typical multiple-neighbor interactions, finding an appropriate boundary condition that models a desired traction is a non-trivial task. We first present a one-dimensional example, which demonstrates how such boundary conditions can be formulated. We further analyze the stability, and derive its continuum limit. We also show how the boundary conditions can be extended to higher dimensions with an application to a dislocation dipole problem under shear stress.Comment: 21 pages; 10 figure