Hoover Dynamics Research Articles

Equilibration in the Nosé–Hoover Isokinetic Ensemble: Effect of Inter-Particle Interactions

We investigate the stationary and dynamic properties of the celebrated Nosé–Hoover dynamics of many-body interacting Hamiltonian systems, with an emphasis on the effect of inter-particle interactions. To this end, we consider a model system with both short- and long-range interactions. The Nosé–Hoover dynamics aim to generate the canonical equilibrium distribution of a system at a desired temperature by employing a set of time-reversible, deterministic equations of motion. A signature of canonical equilibrium is a single-particle momentum distribution that is Gaussian. We find that the equilibrium properties of the system within the Nosé–Hoover dynamics coincides with that within the canonical ensemble. Moreover, starting from out-of-equilibrium initial conditions, the average kinetic energy of the system relaxes to its target value over a size-independent timescale. However, quite surprisingly, our results indicate that under the same conditions and with only long-range interactions present in the system, the momentum distribution relaxes to its Gaussian form in equilibrium over a scale that diverges with the system size. On adding short-range interactions, the relaxation is found to occur over a timescale that has a much weaker dependence on system size. This system-size dependence of the timescale vanishes when only short-range interactions are present in the system. An implication of such an ultra-slow relaxation when only long-range interactions are present in the system is that macroscopic observables other than the average kinetic energy when estimated in the Nosé–Hoover dynamics may take an unusually long time to relax to its canonical equilibrium value. Our work underlines the crucial role that interactions play in deciding the equivalence between Nosé–Hoover and canonical equilibrium.

Adaptive Runge–Kutta integration for stiff systems: Comparing Nosé and Nosé–Hoover dynamics for the harmonic oscillator

We describe the application of adaptive (variable time step) integrators to stiff differential equations encountered in many applications. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely related examples, Nosé's dynamics and Nosé–Hoover dynamics, are both based on Hamiltonian mechanics and generate microstates consistent with Gibbs' canonical ensemble. Nosé's dynamics is stiff and can present severe numerical difficulties. Nosé–Hoover dynamics, although it follows exactly the same trajectory, is smooth and relatively trouble-free. We emphasize the power of adaptive integrators to resolve stiff problems such as the Nosé dynamics for the harmonic oscillator. The solutions also illustrate the power of computer graphics to enrich numerical solutions.

Mesoscopic virial equation for nonequilibrium statistical mechanics

We derive a class of mesoscopic virial equations governing energy partition between conjugate position and momentum variables of individual degrees of freedom. They are shown to apply to a wide range of nonequilibrium steady states with stochastic (Langevin) and deterministic (Nosé–Hoover) dynamics, and to extend to collective modes for models of heat-conducting lattices. A macroscopic virial theorem ensues upon summation over all degrees of freedom. It allows for the derivation of generalised (nonequilibrium) equations of state that involve average dissipative heat flows besides genuine state variables, as exemplified for inertial Brownian motion with solid friction and overdamped active Brownian particles subject to inhomogeneous pressure.

Study of critical dynamics in fluids via molecular dynamics in canonical ensemble.

With the objective of understanding the usefulness of thermostats in the study of dynamic critical phenomena in fluids, we present results for transport properties in a binary Lennard-Jones fluid that exhibits liquid-liquid phase transition. Various collective transport properties, calculated from the molecular dynamics (MD) simulations in canonical ensemble, with different thermostats, are compared with those obtained from MD simulations in microcanonical ensemble. It is observed that the Nosé-Hoover and dissipative particle dynamics thermostats are useful for the calculations of mutual diffusivity and shear viscosity. The Nosé-Hoover thermostat, however, as opposed to the latter, appears inadequate for the study of bulk viscosity.

Prediction of excess pressure drop in contraction–expansion flow by molecular dynamics: Axisymmetric and planar configurations

In this paper, non-equilibrium molecular dynamics is used to simulate the flow of a dilute solution of linear molecules (Boger fluid) and a simple fluid (Lennard-Jones) through axisymmetric and planar contraction–expansion geometries. The pressure flow condition is generated by adding an external force field Fe to the equation of motion for the velocity, which is coupled to Nose–Hoover dynamics to keep the temperature constant. According to the monomer-spring model of Kremer and Grest, linear molecules are represented; and the simple fluid consists of spherical particles, which interact by means of a Lennard-Jones potential. The rheological response of the fluids indicates that the Boger fluid and the simple fluid exhibit constant viscosity in the interval (0.002⩽γ̇⩽0.5); additionally, the Boger fluid presents elastic effects under shear (first normal stress difference, N1) which are quadratic at low shear rates. In pressure flow through the expansion–contraction, results indicate that when both fluids have the same viscosity, pressure profiles P(x1) in the axisymmetric geometry reveal a higher pressure drop (ΔP) in the Boger fluid, while in the planar geometry ΔP was the same for both fluids. Results also reveal that ΔP is closely related to the extensional strain rate (ε̇) experienced by the fluid at the contraction entrance. The pressure drop is higher in the axisymmetric geometry because the change in the molecular conformation, as measured by the mean-square mass distribution tensor 〈I22〉, is 80% higher than in the planar case, resulting in an increase in the energy required to deform the molecule and the loss of mechanical energy. In the planar geometry, under the same extensional strain rate, the conformational change of the molecules in the Boger fluid at the contraction is then lower than in the axisymmetric geometry.

Elastic response from pressure drop measurements through planar contraction–expansion geometries by molecular dynamics: structural effects in melts and molecular origin of excess pressure drop

In this work, we use nonequilibrium molecular dynamics to simulate a contraction–expansion flow of various systems, namely melts with molecules of various conformations (linear, branched, and star), linear molecules in solution, and a reference Lennard–Jones fluid. The equations for Poiseuille flow are solved using a multiple time scale algorithm extended to nonequilibrium situations. Simulations are performed at constant temperature using the Nose–Hoover dynamics. The main objective of this analysis is to investigate the molecular origin of pressure drop along planar contraction–expansion geometry, varying the length of the contraction, and the effect that different molecular conformations have on the resulting pressure drop along the geometry. Pressure drop is closely related to mass distribution (in neutral and gradient directions) and branching index of molecules. Also, it is shown that remarkable increases of pressure drops are also possible in planar geometries, provided large extensional viscosities combined with moderate values of the first normal stress difference in shear are considered, in addition to considerable reductions of the flow area at the contraction region.

Advanced techniques for constrained internal coordinate molecular dynamics

Internal coordinate molecular dynamics (ICMD) methods provide a more natural description of a protein by using bond, angle, and torsional coordinates instead of a Cartesian coordinate representation. Freezing high-frequency bonds and angles in the ICMD model gives rise to constrained ICMD (CICMD) models. There are several theoretical aspects that need to be developed to make the CICMD method robust and widely usable. In this article, we have designed a new framework for (1) initializing velocities for nonindependent CICMD coordinates, (2) efficient computation of center of mass velocity during CICMD simulations, (3) using advanced integrators such as Runge-Kutta, Lobatto, and adaptive CVODE for CICMD simulations, and (4) cancelling out the "flying ice cube effect" that sometimes arises in Nosé-Hoover dynamics. The Generalized Newton-Euler Inverse Mass Operator (GNEIMO) method is an implementation of a CICMD method that we have developed to study protein dynamics. GNEIMO allows for a hierarchy of coarse-grained simulation models based on the ability to rigidly constrain any group of atoms. In this article, we perform tests on the Lobatto and Runge-Kutta integrators to determine optimal simulation parameters. We also implement an adaptive coarse-graining tool using the GNEIMO Python interface. This tool enables the secondary structure-guided "freezing and thawing" of degrees of freedom in the molecule on the fly during molecular dynamics simulations and is shown to fold four proteins to their native topologies. With these advancements, we envision the use of the GNEIMO method in protein structure prediction, structure refinement, and in studying domain motion.

Predictions of the excess pressure drop of Boger fluids through a 2:1:2 contraction–expansion geometry using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamics are used to generate the flow of polymer solutions, specifically of Boger fluids, through a planar 2:1:2 contraction–expansion geometry. The solvent molecules are represented by Lennard–Jones particles, while linear molecules are described by spring-monomers with a finite extensible non-linear elastic spring potential. The equations for Poiseuille flow are solved using a multiple time-scale algorithm extended to non-equilibrium situations. Simulations are performed at constant temperature using Nose–Hoover dynamics. At simulation conditions, changes in concentration show no significant effect on molecular conformation, velocity profiles, and stress fields, while variations in the Deborah number have a strong influence on fluid response. Increasing the magnitude of the Deborah number (De), larger deformation rates are developed in the flow region. For a Deborah number of one, the non-dimensional pressure drop presents values lower than the correspondent Newtonian case. However, for large Deborah numbers, the pressure drop increases above the Newtonian reference. An effective excess pressure drop above the Newtonian value is predicted for Boger fluids along this geometry.

Non-ergodicity of Nosé–Hoover dynamics**This paper is published as part of a collection in honour of Todd Dupont's 65th birthday.

The Nosé–Hoover dynamics is a deterministic method that is commonly used to sample the canonical Gibbs measure. This dynamics extends the physical Hamiltonian dynamics by the addition of a ‘thermostat’ variable, which is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nosé–Hoover dynamics for the one-dimensional harmonic oscillator.In this paper, this result is extended to non-harmonic one-dimensional systems. We also show that, for some multidimensional systems, the averaged dynamics for the limit of infinite thermostat ‘mass’ has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.

Qualitative study of nanocluster positioning process: Planar molecular dynamics simulations

One of the key factors in the assembly of nanoclusters is the precise positioning of them by a manipulation system. Currently the size of clusters used as building blocks is shrinking down to a few nanometers. In such cases, the particle nature of matter plays an important role in the manipulator/cluster/substrate interactions. Having a deeper insight to the aforementioned nano-scale interactions is crucial for prediction and understanding of the behavior of nanoclusters during the positioning process. In the present research, 2D molecular dynamics simulations have been used to investigate such behaviors. Performing planar simulations can provide a fairly acceptable qualitative tool for our purpose while the computation time is greatly reduced in comparison to 3D simulations. The system consists of a tip, cluster and substrate. The focus of the present research is on ultra-fine metallic nanoclusters. To perform this research, Nose–Hoover dynamics and Sutton–Chen interatomic potential will be used to investigate the behavior of the above system which is made from different transition metals. The effects of material type, tip form and manipulation strategy on the success of the process have been investigated by planar molecular dynamics. Such qualitative simulation studies can evaluate the chance of success of a certain nanopositioning scenario regarding different working conditions before consuming large-scale computation time or high experimental expenses.

Markovian Approximation for the Nosé–Hoover Method and H-Theorem

A Langevin equation with state-dependent random force is considered. When the Helmholtz free energy is a nonincreasing function of time (the H-theorem), a generalized Einstein relation is obtained. A stochastic process of the Nose–Hoover method is discussed on the basis of the Markovian approximation. It is found that the generalized Einstein relation holds for the Fokker–Planck equation associated with the stochastic Nose–Hoover equation. The present result indicates that the Nose–Hoover dynamics coarse-grained with time satisfies the H-theorem and therefore the canonical distribution is guaranteed for the subsystem.

Nosè–Hoover dynamics in quantum phase space

Thermal fluctuations in time-dependent quantum processes are treated by a constant-temperature generalization of Wigner's formulation of quantum mechanics in phase space. To this end, quantum Nosè–Hoover dynamics is defined by generalizing the Moyal bracket. Computational applications of the formalism, together with further theoretical developments, are discussed.

Nosé–Hoover nonequilibrium dynamics and statistical mechanics

At equilibrium Nosé's 1984 revolutionary thermostat idea linked Newton's mechanics with Gibbs' statistical mechanics. His work expanded the scope of isothermal and isobaric simulations. Nosé–Hoover dynamics has subsequently facilitated the simulation and detailed understanding of nonequilibrium problems. The fractal phase-space distributions, and their close link to the Lyapunov spectrum, provide a novel explanation of irreversibility and a rich field for exploration.

Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

The Nose–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nose–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nose–Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nose–Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nose–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.

Structure and Dynamics of a Benzenedithiol Monolayer on a Au(111) Surface

We use the universal force field (UFF) developed by Rappe et al. (Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A.; Skiff, W. M. J. Am. Chem. Soc. 1992, 114, 10024) and the specific classical potentials developed from ab initio calculations for Au−benzenedithiol (BDT) molecule interaction to perform molecular dynamics (MD) simulations of a BDT monolayer on an extended Au(111) surface. The simulation system consists of 100 BDT molecules and three rigid Au layers in a simulation box that is rhombic in the plane of the Au surface. A multiple time scale algorithm, the double-reversible reference system propagator algorithm (double RESPA) based on the Nose−Hoover dynamics scheme, and the Ewald summation with a boundary correction term for the treatment of long-range electrostatic interactions in a 2-D slab have been incorporated into the simulation technique. We investigate the local bonding properties of Au−BDT contacts and molecular orientation distributions of BDT molecules. These results show t...

Simulations of crystallization and melting of the FCC (1 0 0) interface: the crucial role of lattice imperfections

We present nonequilibrium simulations of growth and melting of the atomic FCC (1 0 0) interface. Using Nosé–Hoover dynamics we have carefully studied size effects and approximated the dynamics of the solid–liquid interface in a large system as closely as possible. This led to a clear asymmetry of growth and melting rates close to equilibrium. It was possible to explain these findings in terms of the lattice imperfections in crystalline phases in contact with a liquid phase, which automatically developed during growth simulations but were absent in the melting simulations. It was shown that when melting simulations were started with appropriate starting configurations, the asymmetry could be made to disappear.

The Nosé–Poincaré Method for Constant Temperature Molecular Dynamics

We present a new extended phase space method for constant temperature (canonical ensemble) molecular dynamics. Our starting point is the Hamiltonian introduced by Nosé to generate trajectories corresponding to configurations in the canonical ensemble. Using a Poincaré time-transformation, we construct a Hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble. Our approach corrects a serious deficiency of the standard change of variables (Nosé–Hoover dynamics), which yields a time-reversible system but simultaneously destroys the Hamiltonian structure. A symplectic discretization method is presented for solving the Nosé–Poincaré equations. The method is explicit and preserves the time-reversal symmetry. In numerical experiments, it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large. Extensions are presented for Nosé chains, holonomic constraints, and rigid bodies.

Comment on ‘‘Constant pressure molecular dynamics algorithms’’ [J. Chem. Phys. 101, 4177 (1994)

In this comment we point out the differences between the NPT ensemble techniques proposed in references [S. Melchionna, G. Ciccotti, and B. L. Holian, Mol. Phys. 78, 533 (1993)] and [W. G. Hoover, Phys. Rev. A 31, 1695 (1985)], i.e., the revised form and the original form of the Hoover NPT dynamics. We show that our dynamics generates exactly the isobaric–isothermal distribution function for a system with (N−1) particles (similarly to what happens with the molecular dynamic (MD) microcanonical ensemble in which the total momentum is conserved). Moreover we show numerically that in the original Hoover dynamics, i.e., the one leading to an incorrect distribution function, the Nosè variable thermalizes the center-of-mass velocity while that is exactly what does not happen with our dynamics which produces the correct isobaric distribution.

Constant Temperature Constrained Molecular Dynamics: The Newton−Euler Inverse Mass Operator Method

The Newton−Euler inverse mass operator (NEIMO) method for internal coordinate molecular dynamics (MD) of macromolecules (proteins and polymers) leads to stable dynamics for time steps about 10 times larger than conventional dynamics (e.g., 20 or 30 fs rather than 1 or 2 fs for systems containing hydrogens). NEIMO is practical for large systems since the computation time scales linearly with the number of degrees of freedom 𝒩 (instead of the 𝒩 3 scaling for conventional constrained MD methods). In this paper we generalize the NEIMO formalism to the Nose (and Hoover) thermostat to derive the Nose and Hoover equations of motion for constrained canonical ensemble molecular dynamics. We also examined the optimum mass, Q, determining the time scale (τs) for exchange of energy with the heat bath for NEIMO−Hoover dynamics of polymers. We carried out NEIMO−Hoover simulations on the amorphous polymers poly(vinyl chloride) and poly(vinylidene fluoride), where we find that time steps of 20−30 fs lead to stable dynami...

General theory of Onsager symmetries for perturbations of equilibrium and nonequilibrium steady states

The theory of Onsager symmetry is reconsidered from the point of view of its application to nonequilibrium, possibly turbulent steady states. A dynamical formalism based on correlation and response functions is used; understanding of its relationship to more conventional approaches based on entropy production enables one to resolve various confusions about the proper use of the theory, even near thermal equilibrium. Previous claims that ‘‘kinematic’’ flows must be excluded from considerations of Onsager symmetry are refuted by showing that suitably defined reversible and irreversible parts of the Onsager matrix separately obey the appropriate symmetry; fluctuating hydrodynamics serves as an example. It is shown that Onsager symmetries are preserved under arbitrary covariant changes of variables; the Weinhold metric is used as a fundamental tensor. Covariance is used to render moot the controversy over the proper choice of fluxes and forces in neoclassical plasma transport theory. The fundamental distinction between the fully contravariant Onsager matrix Lij and its mixed representation Lij is emphasized and used to explain why some previous workers have failed to find Onsager symmetry around turbulent steady states. The generalized Onsager theorem of Dufty and Rubí [Phys. Rev. A 36, 222 (1987)] is reviewed. An explicitly soluble Langevin problem is shown to violate Onsager’s original symmetry but to obey the generalized theorem. The physical content of the generalized Onsager symmetry is discussed from the point of view of Nosé–Hoover dynamics. A set of extended Graham–Haken potential conditions are derived for Fokker–Planck models and shown to be consistent with the generalized Onsager relations. Finally, for quite general, possibly turbulent steady states it is argued that realizable Markovian statistical closures with underlying Langevin representations must also obey the generalized theorem. In the special case in which all state variables have even parity and there are no external parameters that change sign under time reversal, the steady-state energy balance fully determines the Onsager matrix, which is guaranteed to be symmetric.