A simple, direct derivation and proof of the validity of the SLLOD equations of motion for generalized homogeneous flows

We present a simple and direct derivation of the SLLOD equations of motion for molecular simulations of general homogeneous flows. We show that these equations of motion (1) generate the correct particle trajectories, (2) conserve the total thermal momentum without requiring the center of mass to be located at the origin, and (3) exactly generate the required energy dissipation. These equations of motion are compared with the g-SLLOD and p-SLLOD equations of motion, which are found to be deficient. Claims that the SLLOD equations of motion are incorrect for elongational flows are critically examined and found to be invalid. It is confirmed that the SLLOD equations are, in general, non-Hamiltonian. We derive a Hamiltonian from which they can be obtained in the special case of a symmetric velocity gradient tensor. In this case, it is possible to perform a canonical transformation that results in the well-known DOLLS tensor Hamiltonian.     

Non-Born-Oppenheimer treatment of the H2 Hookean molecule

We show that the exact non-Born-Oppenheimer Schrodinger equation for the Hookean diatomic molecule H2 (a two-proton, two-electron system where the electron-proton interaction is harmonic while the proton-proton and electron-electron interactions are Coulombic) can be decoupled into equations describing the relative motion of the electrons, the relative motion of nuclei, the motion of a collective mode representing a three-dimensional harmonic oscillator, and the motion of a free particle expressed as a linear combination of the individual center-of-mass coordinates of the nuclei and electrons. Analytic solutions to the relative motion of electrons can be readily obtained for the given values of the harmonic coupling constant. However, exact analytic solutions to the equation for the relative motion of the nuclei cannot be obtained simultaneously due to the fact that the harmonic constants in these two equations are coupled. For this reason, we present for the relative nuclear motion approximate analytic wave functions, one of them obtained variationally and the other by a series solution where the coefficients are determined recursively. We also explore a variational solution to the Taylor-series expansion of the nuclear interaction potential. Properties of the electronic and nuclear intracule densities are examined at different values of the coupling constant. An interesting result of the present non-Born-Oppenheimer treatment of this harmonic model is the fact that the relative nuclear motion occurs in a highly correlated regime. This leads in a natural way to a spatial localization of the nuclei akin to Wigner electronic crystallization.     

A validation of the p-SLLOD equations of motion for homogeneous steady-state flows

A validation of the p-SLLOD equations of motion for nonequilibrium molecular dynamics simulation under homogeneous steady-state flow is presented. We demonstrate that these equations generate the correct center-of-mass trajectory of the system, are completely compatible with (and derivable from) Hamiltonian dynamics, satisfy an appropriate energy balance, and require no fictitious external force to generate the required homogeneous flow. It is also shown that no rigorous derivation of the SLLOD equations exists to date.     

Dissipaton equation of motion for system-and-bath interference dynamics

In this review we give a comprehensive account on the dissipaton equation of motion(DEOM) approach to quantum mechanics of open systems. This approach provides a statistical quasi-particle(dissipaton) picture for the environment, as it participates in the correlated system-and-bath dynamics. The underlying dissipaton algebra is de facto established via a close comparison with the celebrated hierarchical equations of motion formalism that is rooted at the Feynman-Vernon influence functional path integral formalism. As a quasi-particle generalization, DEOM identifies unambiguously the physical meanings of all involving dynamical variables as many-dissipaton configurations. It addresses the dynamics of not only systems but also hybridizing bath degrees of freedom. We demonstrate these features of DEOM via its real-time evaluation of the Fano interference of an analytically solvable model system, with the highlight that the statistical quasi-particle picture is ubiquitous, implied even in those commonly used quantum master equations.     

Reversible measure-preserving integrators for non-Hamiltonian systems

We present a systematic method for deriving reversible measure-preserving integrators for non-Hamiltonian systems such as the Nosé-Hoover thermostat and generalized Gaussian moment thermostat (GGMT). Our approach exploits the (non-Poisson) bracket structure underlying the thermostat equations of motion. Numerical implementation for the GGMT system shows that our algorithm accurately conserves the thermostat energy function. We also study position and momentum distribution functions obtained using our integrator.     

A preliminary study on sea wave packet equations on slowly varying topography

There is a common hypothesis for the presently popular mild-slope equations that wave particle motion is irrotational. In this paper, an attempt is made to abandon the irrotational assumption and to set up new sea wave packet equations on slowly varying topography by use of the WKBJ method. To simplify the deduction, the two-dimensional shallow water equations are used to describe the sea wave particle motion in the very shallow nearshore area. The established equations can give some characteristics of wave propagation near shore.     

A preliminary study on sea wave packet equations on slowly varying topography

There is a common hypothesis for the presently popular mild-slope equations that wave particle motion is irrotational. In this paper, an attempt is made to abandon the irrotational assumption and to set up new sea wave packet equations on slowly varying topography by use of the WKBJ method. To simplify the deduction, the two-dimensional shallow water equations are used to describe the sea wave particle motion in the very shallow nearshore area. The established equations can give some characteristics of wave propagation near shore.     

Modal dynamics of proteins in water

We studied a dynamical model for the motion of the large scales of proteins in water. The model was obtained by projecting the (averaged) Newton equations onto some set of harmonic modes. We compared the statistics of the so‐obtained trajectories with those obtained by standard techniques, and concluded that our dynamical model is able to fairly reproduce the average properties of the large‐scale motion of the protein, and at the same time allow time steps one order of magnitude larger than the standard ones. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1274–1282, 2000     

Natural convection in nanofluid flow with chemotaxis process over a vertically inclined heated surface

The thermal energy transport analysis with chemotaxis in the free convective flow of viscous nanofluid over stretchable vertically inclined heated sheet is addressed in this article. The fluid forced and free convection motion is investigated and discussed with physical reasoning. The fluid also contains microorganism heavy-bottom species, and their chemotactic motion is studied. In the light of Buongiorno model, the impact of Brownian motion and thermophoresis slip mechanism on thermal conduction in the nanofluid is analyzed. The work is based on the similarity analysis of governing partial differential equations (PDEs) which lead to non-dimensional ordinary differential equations (ODEs). The solution of resulting flow and heat equations is computed via bvp4c technique. The outcomes are represented in graphical abstract. It is noted that free convective flow field increases near to the surface of sheet then it decays to free stream exponentially. Higher magnitude of thermophoretic force boost up the thermal energy transport in nanofluid flow. The Brownian motion enhances temperature profile and lower down the convection velocity. Chemotaxis motion of species in nanofluid is increasing function of bioconvective Peclet number.     

Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.     

Compact adaptive-grid scheme for high numerical resolution simulations of isotachophoresis

In a previous publication we demonstrated a fast simulation tool for solution of electrophoretic focusing and separation. We here describe the novel mathematical model and numerical algorithms used to create this code. These include the representation of advection–diffusion equations on an adaptive grid, high-resolution discretization of the equations (sixth order compact), a new variational-based approach for controlling the motion of grid points, and new boundary conditions which enable solution in a moving frame of reference. We discuss the advantages of combining a high-resolution discretization with an adaptive grid in accurately resolving sharp interfaces in isotachophoresis, and provide verification against known analytical solutions and comparison with prevailing exiting numerical algorithms.     

Operator splitting algorithm for isokinetic SLLOD molecular dynamics

We apply an operator splitting method to develop a simulation algorithm that has complete analytical solutions for the Gaussian thermostated SLLOD equations of motion [D. J. Evans and G. P. Morriss, Phys. Rev. A 30, 1528 (1984)] for a system under shear. This leads to a homogeneous algorithm for performing both equilibrium and nonequilibrium isokinetic molecular dynamics simulation. The resulting algorithm is computationally efficient. In particular, larger integration time steps can be used compared to simulations with regular Gaussian thermostated SLLOD equations of motion. The utility and accuracy of the algorithm are demonstrated through application to the Weeks-Chandler-Anderson fluid. Although strict conservation of the kinetic energy suppresses thermal fluctuations in the system, this algorithm does not allow simulations at lower shear rates than those normally afforded by older nonequilibrium molecular dynamics simulations.     

Non-adiabatic quantum molecular dynamics: basic formalism and case study

A general formalism is presented that treats selfconsistently and simultaneously classical atomic motion and quantum electronic excitations in dynamical processes of atomic many-body systems (non-adiabatic quantum molecular dynamics). On the basis of time-dependent density functional theory, coupled highly non-linear equations of motion are derived for arbitrary basis sets for the time-dependent Kohn-Sham orbitals. Possible approximations to make the approach practical for large atomic cluster systems are discussed. As a first application of the still exact equations of motion, non-adiabatic effects in the scattering of H⁺+H, as a case study, are investigated.     

Optimizing hierarchical equations of motion for quantum dissipation and quantifying quantum bath effects on quantum transfer mechanisms

We present an optimized hierarchical equations of motion theory for quantum dissipation in multiple Brownian oscillators bath environment, followed by a mechanistic study on a model donor-bridge-acceptor system. We show that the optimal hierarchy construction, via the memory-frequency decomposition for any specified Brownian oscillators bath, is generally achievable through a universal pre-screening search. The algorithm goes by identifying the candidates for the best be just some selected Padé spectrum decomposition based schemes, together with a priori accuracy control criterions on the sole approximation, the white-noise residue ansatz, involved in the hierarchical construction. Beside the universal screening search, we also analytically identify the best for the case of Drude dissipation and that for the Brownian oscillators environment without strongly underdamped bath vibrations. For the mechanistic study, we quantify the quantum nature of bath influence and further address the issue of localization versus delocalization. Proposed are a reduced system entropy measure and a state-resolved constructive versus destructive interference measure. Their performances on quantifying the correlated system-environment coherence are exemplified in conjunction with the optimized hierarchical equations of motion evaluation of the model system dynamics, at some representing bath parameters and temperatures. Analysis also reveals the localization to delocalization transition as temperature decreases.     

Multipartite quantum entanglement evolution in photosynthetic complexes

We investigate the evolution of entanglement in the Fenna-Matthew-Olson (FMO) complex based on simulations using the scaled hierarchical equations of motion approach. We examine the role of entanglement in the FMO complex by direct computation of the convex roof. We use monogamy to give a lower bound for entanglement and obtain an upper bound from the evaluation of the convex roof. Examination of bipartite measures for all possible bipartitions provides a complete picture of the multipartite entanglement. Our results support the hypothesis that entanglement is maximum primary along the two distinct electronic energy transfer pathways. In addition, we note that the structure of multipartite entanglement is quite simple, suggesting that there are constraints on the mixed state entanglement beyond those due to monogamy.     

Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions

We present a lattice Boltzmann solution of the equations of motion describing the spreading of droplets on topologically patterned substrates. We apply it to model superhydrophobic behavior on surfaces covered by an array of micrometer-scale posts. We find that the patterning results in a substantial increase in contact angle, from 110 degrees to 156 degrees. The dynamics of the transition from drops suspended on top of the posts to drops collapsed in the grooves is described.     

Time propagation of constrained coupled Gaussian wave packets

The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.     

A configurational temperature Nosé-Hoover thermostat

We propose two new thermostats which can be employed in computer simulations to ensure that two different variants of the configurational temperature fluctuate around their equilibrium values. These new thermostats differ from one previously introduced by Delhommelle and Evans [Mol. Phys. 99, 1825 (2001)] in several important ways. First, our thermostats are derived in the same spirit as the Nosé-Hoover thermostat and therefore generate the canonical phase-space distribution. Second, our thermostats involve simpler equations of motion, which do not involve spatial gradients of the configurational temperature. They do not suffer from problems stemming from stiff equations of motion and furthermore, in large temperature perturbation simulations, the measured temperature follows the set-point temperature without any overshoot, and with good damping of oscillations. We show that both of our configurational thermostats are special cases of a more general set of Nosé-Hoover equations proposed by Kusnezov et al. [Ann. Phys. 204, 155 (1990)]. The new thermostats are expected to be highly useful in nonequilibrium simulations, particularly those characterized by spatial inhomogeneities. They should also find applicability in simulations involving large changes in temperature over small time scales, such as temperature quench molecular dynamics and radiation damage modeling.