Temperature control algorithms in dual control volume grand canonical molecular dynamics simulations of hydrogen diffusion in palladium

The effectiveness of five temperature control algorithms for dual control volume grand canonical molecular dynamics is investigated in the study of hydrogen atom diffusion in a palladium bulk. The five algorithms, namely, Gaussian, generalized Gaussian moment thermostat (GGMT), velocity scaling, Nosé-Hoover (NH), and its enhanced version Nosé-Hoover chain (NHC) are examined in both equilibrium and nonequilibrium simulation studies. Our numerical results show that Gaussian yields the most inaccurate solutions for the hydrogen-palladium system due to the high friction coefficient generated from the large velocity fluctuation of hydrogen, while NHC, NH, and GGMT produce the most accurate temperature and density profiles in both equilibrium and nonequilibrium cases with their feedback control mechanisms. However, this feedback control also overestimates the self-diffusion coefficients in equilibrium systems and the diffusion coefficient in nonequilibrium systems. Velocity scaling thermostat produces slight inhomogeneities in the temperature and density profiles, but due to the dissipated heat accumulated in the control volumes it still yields accurate self-diffusion coefficients that are in good agreement with the experimental data at a wide range of temperatures while others tend to deviate.     

From molecular dynamics to hydrodynamics: a novel Galilean invariant thermostat

This paper proposes a novel thermostat applicable to any particle-based dynamic simulation. Each pair of particles is thermostated either (with probability P) with a pairwise Lowe-Andersen thermostat [C. P. Lowe, Europhys. Lett. 47, 145 (1999)] or (with probability 1-P) with a thermostat that is introduced here, which is based on a pairwise interaction similar to the Nosé-Hoover thermostat. When the pairwise Nosé-Hoover thermostat dominates (low P), the liquid has a high diffusion coefficient and low viscosity, but when the Lowe-Andersen thermostat dominates, the diffusion coefficient is low and viscosity is high. This novel Nosé-Hoover-Lowe-Andersen thermostat is Galilean invariant and preserves both total linear and angular momentum of the system, due to the fact that the thermostatic forces between each pair of the particles are pairwise additive and central. We show by simulation that this thermostat also preserves hydrodynamics. For the (noninteracting) ideal gas at P = 0, the diffusion coefficient diverges and viscosity is zero, while for P > 0 it has a finite value. By adjusting probability P, the Schmidt number can be varied by orders of magnitude. The temperature deviation from the required value is at least an order of magnitude smaller than in dissipative particle dynamics (DPD), while the equilibrium properties of the system are very well reproduced. The thermostat is easy to implement and offers a computational efficiency better than (DPD), with better temperature control and greater flexibility in terms of adjusting the diffusion coefficient and viscosity of the simulated system. Applications of this thermostat include all standard molecular dynamic simulations of dense liquids and solids with any type of force field, as well as hydrodynamic simulation of multiphase systems with largely different bulk viscosities, including surface viscosity, and of dilute gases and plasmas.     

Multiple time scale molecular dynamics for fluids with orientational degrees of freedom. II. Canonical and isokinetic ensembles

We have developed several multiple time stepping techniques to overcome the limitations on efficiency of molecular dynamics simulations of complex fluids. They include the modified canonical and isokinetic schemes, as well as the extended isokinetic Nosé-Hoover chain approach. The latter generalizes the method of Minary, Tuckerman, and Martyna for translational motion [Phys. Rev. Lett. 93, 150201 (2004)] to systems with both translational and orientational degrees of freedom. Although the microcanonical integrators are restricted to relatively small outer time steps of order of 16 fs, we show on the basis of molecular dynamics simulations of ambient water that in the canonical and isokinetic thermostats the size of these steps can be increased to 50 and 75 fs, respectively (at the same inner time step of 4 fs). Within the generalized isokinetic Nosé-Hoover chain algorithm we have derived, huge outer time steps of order of 500 fs can be used without losing numerical stability and affecting equilibrium properties.     

Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules

Molecular dynamics integrators are presented for translational and rotational motion of rigid molecules in microcanonical, canonical, and isothermal-isobaric ensembles. The integrators are all time reversible and are also, in some approaches, symplectic for the microcanonical ensembles. They are developed utilizing the quaternion representation on the basis of the Trotter factorization scheme using a Hamiltonian formalism. The structure is similar to that of the velocity Verlet algorithm. Comparison is made with standard integrators in terms of stability and it is found that a larger time step is stable with the new integrators. The canonical and isothermal-isobaric molecular dynamics simulations are defined by using a chain thermostat approach according to generalized Nosé-Hoover and Andersen methods.     

Efficient stochastic thermostatting of path integral molecular dynamics

The path integral molecular dynamics (PIMD) method provides a convenient way to compute the quantum mechanical structural and thermodynamic properties of condensed phase systems at the expense of introducing an additional set of high frequency normal modes on top of the physical vibrations of the system. Efficiently sampling such a wide range of frequencies provides a considerable thermostatting challenge. Here we introduce a simple stochastic path integral Langevin equation (PILE) thermostat which exploits an analytic knowledge of the free path integral normal mode frequencies. We also apply a recently developed colored noise thermostat based on a generalized Langevin equation (GLE), which automatically achieves a similar, frequency-optimized sampling. The sampling efficiencies of these thermostats are compared with that of the more conventional Nosé-Hoover chain (NHC) thermostat for a number of physically relevant properties of the liquid water and hydrogen-in-palladium systems. In nearly every case, the new PILE thermostat is found to perform just as well as the NHC thermostat while allowing for a computationally more efficient implementation. The GLE thermostat also proves to be very robust delivering a near-optimum sampling efficiency in all of the cases considered. We suspect that these simple stochastic thermostats will therefore find useful application in many future PIMD simulations.     

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.     

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.     

Configurational temperature control for atomic and molecular systems

A new configurational temperature thermostat suitable for molecules with holonomic constraints is derived. This thermostat has a simple set of motion equations, can generate the canonical ensemble in both position and momentum space, acts homogeneously through the spatial coordinates, and does not intrinsically violate the constraints. Our new configurational thermostat is closely related to the kinetic temperature Nosé-Hoover thermostat with feedback coupled to the position variables via a term proportional to the net molecular force. We validate the thermostat by comparing equilibrium static and dynamic quantities for a fluid of n-decane molecules under configurational and kinetic temperature control. Practical aspects concerning the implementation of the new thermostat in a MOLECULAR DYNAMICS code and the potential applications are discussed.     

The Nosé-Hoover Thermostat in Molecular Dynamics for Nuclear Matter

In this work we present a calculation of the hamiltonian variables solving the molecular dynamics equations of motion for a system of nuclear matter kept at fixed temperature by using the Nosé-Hoover Thermostat and interacting via a semiclassical potential depending on both positions and momenta.     

Crooks equation for steered molecular dynamics using a Nosé-Hoover thermostat

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)], originally derived for microscopically reversible Markovian systems, relates the work done on a system during an irreversible transformation to the free energy difference between the final and the initial state of the transformation. In the present work we provide a theoretical proof of the Crooks equation in the context of constant volume, constant temperature steered molecular dynamics simulations of systems thermostated by means of the Nosé-Hoover method (and its variant using a chain of thermostats). As a numerical test we use the folding and unfolding processes of decaalanine in vacuo at finite temperature. We show that the distribution of the irreversible work for the folding process is markedly non-Gaussian thereby implying, according to Crooks equation, that also the work distribution of the unfolding process must be inherently non-Gaussian. The clearly asymmetric behavior of the forward and backward irreversible work distributions is a signature of a non-Markovian regime for the folding/unfolding of decaalanine.     

Path integral molecular dynamics methods: Application to neon

Feynman's path integral formulation of quantum statistical mechanics, which has commonly been applied be Monte Carlo methods, is now also implemented by traditional molecular dynamics simulations of the microcanonical ensemble and in the Nosé-Hoover method simulating the isothermal-isobaric ensemble. In this article these two methods are applied to solid and liquid neon, in which quantum effects are not negligible. The validity of the procedure is shown by comparison with Monte Carlo and Brownian Dynamics computer simulations and with experiment. © 1995 by John Wiley & Sons, Inc.     

Fully flexible unit cell simulation with recursive thermostat chains

The recursive thermostat chained fully flexible cell molecular dynamic simulation (NsigmaT ensemble) is performed. The ensemble is based on the metric tensor, whose components are used as extended variables. These variables are combined with Nosé-Poincaré recursive thermostat chains. This extended Hamiltonian approach preserves Hamiltonian in structure, and the partition function satisfies the NsigmaT ensemble state in phase space. In the present study, the generalized leap frog method was employed for time integration. The resulting molecular dynamics simulation was performed for bulk and thin film solid materials in the face-centered-cubic crystal structure. Uniaxial tension test and simple shear test are performed to predict the behaviors of a solid material in the bulk state and nanoscale thin film state. The proposed flexible cell method should serve as a powerful tool for the prediction of mechanical and thermal properties of solid materials including nanoscale behavior.     

Understanding Molecular Dynamics with Stochastic Processes via Real or Virtual Dynamics

Molecular dynamics with the stochastic process provides a convenient way to compute structural and thermodynamic properties of chemical, biological, and materials systems. It is demonstrated that the virtual dynamics case that we proposed for the Langevin equation[J. Chem. Phys. 147 , 184104 (2017)] in principle exists in other types of stochastic thermostats as well. The recommended "middle" scheme[J. Chem. Phys. 147 , 034109 (2017)] of the Andersen thermostat is investigated as an example. As shown by both analytic and numerical results, while the real and virtual dynamics cases approach the same plateau of the characteristic correlation time in the high collision frequency limit, the accuracy and efficiency of sampling are relatively insensitive to the value of the collision frequency in a broad range. After we compare the behaviors of the Andersen thermostat to those of Langevin dynamics, a heuristic schematic representation is proposed for understanding efficient stochastic thermostatting processes with molecular dynamics.     

Influence of temperature, friction, and random forces on folding of the B-domain of staphylococcal protein A: all-atom molecular dynamics in implicit solvent

The influences of temperature, friction, and random forces on the folding of protein A have been analyzed. A series of all-atom molecular dynamics folding simulations with the Amber ff99 potential and Generalized Born solvation, starting from the fully extended chain, were carried out for temperatures from 300 to 500 K, using (a) the Berendsen thermostat (with no explicit friction or random forces) and (b) Langevin dynamics (with friction and stochastic forces explicitly present in the system). The simulation temperature influences the relative time scale of the major events on the folding pathways of protein A. At lower temperatures, helix 2 folds significantly later than helices 1 and 3. However, with increasing temperature, the folding time of helix 2 approaches the folding times of helices 1 and 3. At lower temperatures, the complete formation of secondary and tertiary structure is significantly separated in time whereas, at higher temperatures, they occur simultaneously. These results suggest that some earlier experimental and theoretical observations of folding events, e.g., the order of helix formation, could depend on the temperature used in those studies. Therefore, the differences in temperature used could be one of the reasons for the discrepancies among published experimental and computational studies of the folding of protein A. Friction and random forces do not change the folding pathway that was observed in the simulations with the Berendsen thermostat, but their explicit presence in the system extends the folding time of protein A.     

Conserving the linear momentum in stochastic dynamics: Dissipative particle dynamics as a general strategy to achieve local thermostatization in molecular dynamics simulations

Stochastic dynamics is a widely employed strategy to achieve local thermostatization in molecular dynamics simulation studies; however, it suffers from an inherent violation of momentum conservation. Although this short‐coming has little impact on structural and short‐time dynamic properties, it can be shown that dynamics in the long‐time limit such as diffusion is strongly dependent on the respective thermostat setting. Application of the methodically similar dissipative particle dynamics (DPD) provides a simple, effective strategy to ensure the advantages of local, stochastic thermostatization while at the same time the linear momentum of the system remains conserved. In this work, the key parameters to employ the DPD thermostats in the framework of periodic boundary conditions are investigated, in particular the dependence of the system properties on the size of the DPD‐region as well as the treatment of forces near the cutoff. Structural and dynamical data for light and heavy water as well as a Lennard–Jones fluid have been compared to simulations executed via stochastic dynamics as well as via use of the widely employed Nose–Hoover chain and Berendsen thermostats. It is demonstrated that a small size of the DPD region is sufficient to achieve local thermalization, while at the same time artifacts in the self‐diffusion characteristic for stochastic dynamics are eliminated. © 2016 Wiley Periodicals, Inc.     

A global-update simulation method for polymer systems

We describe a hybrid Monte Carlo algorithm, combining molecular dynamics with the Monte Carlo method, applied to the simulation of polymer systems. It is shown that observables are independent of the discretization error, and the performance, i.e., autocorrelation times of observables, is analyzed. For a dense system representing a polyethylene melt, we present data for the pair distribution function and the mean square displacement of a chain in the bulk. We also investigate the possibility of using a scaled Hamiltonian in the algorithm.     

On the correspondence between classical and quantum mechanics in macromolecular systems: Too much classical chaos

In a recent study we found the classical dynamics of a polyethylene (PE) chain to exhibit low dimensional chaos at temperatures as low as a few Kelvin. These results strongly suggest that classical molecular dynamic simulations in polymer systems can grossly overestimate vibrational motion, which consequently results in disordered structures. In contrast, quantum mechanical calculations using Internal Coordinate Quantum Monte Carlo (an improved method with an initial conjecture for the correct wave function) indicate that the quantum ground state for a three-dimensional model PE chain is far more rigid than determined from molecular dynamics (MD) simulations, even at energies as low as a small fraction of the ground state energy. This result casts uncertainty on the reliability of MD estimates of dynamical or structural quantities relevant to the study of some macromolecular systems.     

Monte Carlo simulation and molecular theory of tethered polyelectrolytes

We investigate the structure of end-tethered polyelectrolytes using Monte Carlo simulations and molecular theory. In the Monte Carlo calculations we explicitly take into account counterions and polymer configurations and calculate electrostatic interaction using Ewald summation. Rosenbluth biasing, distance biasing, and the use of a lattice are all used to speed up Monte Carlo calculation, enabling the efficient simulation of the polyelectrolyte layer. The molecular theory explicitly incorporates the chain conformations and the possibility of counterion condensation. Using both Monte Carlo simulation and theory, we examine the effect of grafting density, surface charge density, charge strength, and polymer chain length on the distribution of the polyelectrolyte monomers and counterions. For all grafting densities examined, a sharp decrease in brush height is observed in the strongly charged regime using both Monte Carlo simulation and theory. The decrease in layer thickness is due to counterion condensation within the layer. The height of the polymer layer increases slightly upon charging the grafting surface. The molecular theory describes the structure of the polyelectrolyte layer well in all the different regimes that we have studied.     

Replica exchange and expanded ensemble simulations as Gibbs sampling: simple improvements for enhanced mixing

The widespread popularity of replica exchange and expanded ensemble algorithms for simulating complex molecular systems in chemistry and biophysics has generated much interest in discovering new ways to enhance the phase space mixing of these protocols in order to improve sampling of uncorrelated configurations. Here, we demonstrate how both of these classes of algorithms can be considered as special cases of Gibbs sampling within a Markov chain Monte Carlo framework. Gibbs sampling is a well-studied scheme in the field of statistical inference in which different random variables are alternately updated from conditional distributions. While the update of the conformational degrees of freedom by Metropolis Monte Carlo or molecular dynamics unavoidably generates correlated samples, we show how judicious updating of the thermodynamic state indices--corresponding to thermodynamic parameters such as temperature or alchemical coupling variables--can substantially increase mixing while still sampling from the desired distributions. We show how state update methods in common use can lead to suboptimal mixing, and present some simple, inexpensive alternatives that can increase mixing of the overall Markov chain, reducing simulation times necessary to obtain estimates of the desired precision. These improved schemes are demonstrated for several common applications, including an alchemical expanded ensemble simulation, parallel tempering, and multidimensional replica exchange umbrella sampling.