Nose-Hoover Dynamics Research Articles

Equilibrium free-energy differences at different temperatures from a single set of nonequilibrium transitions.

Crook's fluctuation theorem (CFT) and Jarzynski equality (JE) are effective tools for obtaining free-energy difference ΔF(λ_{A}→λ_{B},T_{0}) through a set of finite-time protocol driven nonequilibrium transitions between two equilibrium states A and B [parametrized by the time-varying protocol λ(t)] at the same temperature T_{0}. Using the generalized dimensionless work function ΔW_{G}, we extend CFT to transitions between two nonequilibrium steady states (NESSs) created by a thermal gradient. We show that it is possible, provided the period over which the transitions occur is sufficiently long, to obtain ΔF(λ_{A}→λ_{B},T_{0}) for different values of T_{0}, using the same set of finite-time transitions between these two NESSs. Our approach thus completely eliminates the need to make new samples for each new T_{0}. The generalized form of JE arises naturally as the average of the exponentiated ΔW_{G}. The results are demonstrated on two test cases: (i) a single particle quartic oscillator having a known closed form ΔF, and (ii) a one-dimensional ϕ^{4} chain. Each system is sampled from the canonical distribution at an arbitrary T^{'} with λ=λ_{A}, then subjected to a temperature gradient between its ends, and after steady state is reached, the protocol change λ_{A}→λ_{B} is effected in time τ, following which ΔW_{G} is computed. The reverse path likewise initiates in equilibrium at T^{'} with λ=λ_{B} and the protocol is time reversed leading to λ=λ_{A} and the reverse ΔW_{G}. Our method is found to be more efficient than either JE or CFT when free-energy differences at multiple T_{0}'s are required for the same system.

Nonergodicity of the Nose-Hoover chain thermostat in computationally achievable time.

The widely used Nose-Hoover chain (NHC) thermostat in molecular dynamics simulations is generally believed to impart the canonical distribution as well as quasi- (i.e., space-filling) ergodicity on the thermostatted physical system (PS). Working with the standard single harmonic oscillator, we prove analytically that the two-chain Nose-Hoover thermostat with unequal thermostat masses approaches the standard Nose-Hoover dynamics, and hence the PS loses its canonical and quasiergodic nature. We also show through numerical simulations over substantially long times that for certain Poincaré sections, for both the equal and unequal thermostat mass cases, the bivariate distribution function of position and momentum (x,p) and of reservoir degrees of freedom (ξ,η) lose their Gaussian nature. Further, the four-dimensional x-p-ξ-η extended phase space exhibits two holes of nonzero measure. The NHC thermostat therefore does not generate the canonical distribution or preserve quasiergodicity for the PS.

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.

Stochastic-dynamical thermostats for constraints and stiff restraints

A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nose-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermostats in the setting of simulating systems with holonomic constraints. The approaches described are easily implemented and facilitate the recovery of correct canonical averages with minimal disturbance of the underlying dynamics. For the purpose of illustrating our results, we examine the numerical application of these methods to a simple atomic chain, where a Fixman term is required to correct the thermodynamic ensemble.

Nonequilibrium Thermodynamics and Nose−Hoover Dynamics

We show that systems driven by an external force and described by Nose-Hoover dynamics allow for a consistent nonequilibrium thermodynamics description when the thermostatted variable is initially assumed in a state of canonical equilibrium. By treating the "real" variables as the system and the thermostatted variable as the reservoir, we establish the first and second law of thermodynamics. As for Hamiltonian systems, the entropy production can be expressed as a relative entropy measuring the system-reservoir correlations established during the dynamics.

Planar molecular dynamics simulation of Au clusters in pushing process

Based on the fact that the manipulation of fine nanoclusters calls for more precise modelling, the aim of this paper is to conduct an atomistic investigation for interaction analysis of particle-substrate system for pushing and positioning purposes. In the present research, 2D molecular dynamics simulations have been used to investigate such behaviours. Performing the planar simulations can provide a fairly acceptable qualitative tool for our purpose while the computation time is reduced extremely in comparison to 3D simulations. To perform this study, Nose-Hoover dynamics and Sutton-Chen interatomic potential will be used to investigate the behaviour of the aforementioned system. Pushing of Au clusters on Au substrate has been chosen as illustrative examples. Dependency of the aforementioned behaviour on temperature has been investigated. Higher temperature affects the pushing success level considerably. In addition, the simulation was performed for Ag cluster as well to compare the performance to one fo...

GNEIMO: Constrained Molecular Dynamics Methods For Long Time Scale Simulation of Macromolecules

The two leading causes for the limitation of the all-atom molecular dynamics (MD) simulation timescales are: 1. calculation of forces that scale as the square of the number of atoms and, 2. the integration time step is limited to 1fs due to the high frequency modes in the protein. High performance technologies and better force calculation algorithms have addressed the former, and we address the latter issue in this work.Here we report a constrained MD method, GNEIMO (Generalized Newton-Euler inverse mass operator method), that is capable of achieving stable dynamics with integration time steps as large as 10 to 20fs. The GNEIMO method provides a platform to perform long time scale hierarchical simulations ranging from all-atom simulations, coarse-grained dynamics of clusters of few atoms, to dynamics with larger motifs constrained, at lesser computational expense compared to all-atom MD. GNEIMO method uses spatial operator algebra to solve for the internal coordinate dynamics with computational cost scaling linearly as the number of degrees of freedom.The current implementation of GNEIMO is capable of performing constant temperature Nose-Hoover dynamics, with continuum Generalized Born solvation. We use adaptive step size integration to provide stable dynamics with larger time steps. We have carried out tens of nano-seconds of stable dynamics for proteins with time steps as large as 10 to 20fs for different integrators. We report results from conformational changes of domains in proteins from long time scale dynamics. This implementation integrates the force-field module from the MD program LAMMPS, with constrained dynamics module from NASA-Jet Propulsion laboratory.

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.

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.

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.

A proof of Jarzynski’s nonequilibrium work theorem for dynamical systems that conserve the canonical distribution

We present a derivation of the Jarzynski [Phys. Rev. Lett. 78, 2690 (1997)] identity and the Crooks [J. Stat. Phys. 90, 1481 (1998)] fluctuation theorem for systems governed by deterministic dynamics that conserves the canonical distribution such as Hamiltonian dynamics, Nose-Hoover dynamics, Nose-Hoover chains, and Gaussian isokinetic dynamics. The proof is based on a relation between the heat absorbed by the system during the nonequilibrium process and the Jacobian of the phase flow generated by the dynamics.

Equilibrium and nonequilibrium states in thermostated gledzer-ohkitani-yamada shell models

We study Gledzer-Ohkitani-Yamada shell models with heat reservoirs. Time-reversible Nose-Hoover dynamics are used for the heat reservoirs. Both equilibrium and nonequilibrium cascade states are obtained by changing the heat reservoirs.

Constrained systems and statistical distribution

Theoretical advances in the statistical description of non-Hamiltonian systems [Tuckerman, Mundy, and Martyna, Europhys. Lett. 45, 149 (1999)] have recently led us to rethink the definition of the phase-space probability by using the proper invariant measure. Starting from this point of view, we will derive the statistical distribution of constrained systems, considered as non-Hamiltonian, in Cartesian coordinates, in a way independent from the standard Lagrangian treatment. Furthermore, we will analyze the statistical distribution of the Nose-Hoover isothermal (canonical or NVT) dynamics, considering with care the conservation laws hidden in the evolution equations. Consequently, we will correct the equations of motion in order to obtain the proper statistical ensemble. The isobaric-isothermal (constant pressure or NpT) form of the Nose-Hoover dynamics is then considered as a nontrivial extension of the described procedure and, similarly to the NVT case, we will correct the equations of motion with respect to previous versions. The case of a constrained system coupled to a thermostat and a piston is easily handled by means of the same formalism, whereas the Lagrangian treatment becomes involved.

Nose-Hoover chain method for nonequilibrium molecular dynamics simulation

It is demonstrated that the Nose-Hoover chain (NHC) thermostat method influences the properties of a system out of equilibrium. A simple modification of the NHC equations is proposed which enables both steady-state averages and time correlation functions to be obtained correctly for nonequilibrium states, and consistent with those computed using Gaussian isokinetic and Nose-Hoover dynamics.

Time-reversible equilibrium and nonequilibrium isothermal-isobaric simulations with centered-difference Stoermer algorithms.

We describe simple, stable, and economical numerical methods for implementing temperature and pressure controls in equilibrium and nonequilibrium Nose-Hoover dynamics. These methods substantially increase the efficiency of large-scale simulations.