A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems : Comparing the Nosé and Nosé-Hoover Oscillator Dynamics
William Graham Hoover, Julien Clinton Sprott, Carol Griswold Hoover
备注：
24 pages with ten figures written for the American Journal of Physics --- Both here and in the published AJP version of October 2016 the time ranges shown in Figure 3 were permuted. 0 to 200 corresponds to the upper portion and 750 to 790 to the lower portion. We regret the error
"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" [ variable timestep ] integrators to "stiff" mechanical problems encountered in modern applications of Gibbs' 1902 statistical mechanics. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely-related examples, Nosé's 1984 dynamics and Nosé-Hoover 1985 dynamics, are both based on Hamiltonian mechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Both these dynamics are consistent with Gibbs' canonical ensemble. Nosé's dynamics is "stiff" and can present severe numerical difficulties. Nosé-Hoover dynamics, though it follows exactly the same trajectory, is "smooth" and relatively trouble-free. Our tutorial emphasises the power of adaptive integrators to resolve stiff problems like the Nosé oscillator. The solutions obtained illustrate the power of computer graphics to enrich numerical solutions. Adaptive integration with computer graphics are basic to an understanding of dynamical systems and statistical mechanics. These tools lead naturally into the visualization of intricate fractal structures formed by chaos as well as elaborate knots tied by regular nonchaotic dynamics. This work was invited by the American Journal of Physics.
We use nonequilibrium molecular dynamics to analyze and illustrate the qualitative differences between the one-thermostat and two-thermostat versions of equilibrium and nonequilibrium (heat-conducting) harmonic oscillators. Conservative nonconducting regions can coexist with dissipative heat conducting regions in phase space with exactly the same imposed temperature field.
Some paradoxical aspects of the Nosé and Nosé-Hoover dynamics of 1984 and Dettmann's dynamics of 1996 are elucidated. Phase-space descriptions of thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, as is described here by a variety of three- and four-dimensional phase-space models. These findings illustrate some surprising consequences when Liouville's continuity equation is applied to Hamiltonian flows.
To follow up recent work of Xiao-Song Yang on the Nosé-Hoover oscillator we consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator to relate our mechanical studies to Gibbs' statistical mechanics. All three oscillators are described by a coordinate $q$ and a momentum $p$. Additional control variables $(ζ, ξ)$ vary the energy. Dettmann's description includes a time-scaling variable $s$, as does Nosé's original work. Time scaling controls the rates at which the $(q,p,ζ)$ variables change. The ergodic Hoover-…
The popular method of Nose and Hoover to create canonically distributed positions and momenta in classical molecular dynamics simulations is generalized to a genuine quantum system of infinite dimensionality. We show that for the quantum harmonic oscillator, the equations of motion in terms of coherent states can easily be modified in an analogous manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble. Possible applications to more complex systems, especially interacting Fermion systems, are proposed.
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 Nosé--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 Nosé--Hoover equation. The present result indicates that the Nosé--Hoover dynamics coarse-grained with time satisfies the H-theorem and therefore works as a heat bath.
We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nosé-Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force, in the limit where it is small.
Aspects of the Nosé and Nosé-Hoover dynamics developed in 1983-1984 along with Dettmann's closely related dynamics of 1996, are considered. We emphasize paradoxes associated with Liouville's Theorem. Our account is pedagogical, focused on the harmonic oscillator for simplicity, though exactly the same ideas can be, and have been, applied to manybody systems. Nosé, Nosé-Hoover, and Dettmann flows were all developed in order to access Gibbs' canonical ensemble directly from molecular dynamics. Unlike Monte Carlo algorithms dynamical flow models are often not ergodic and so can fail to reproduce …
We compare simulations performed using the Nosé-Hoover and the Langevin thermostats with the Hamiltonian dynamics of a long-range interacting system in contact with a reservoir. We find that while the statistical mechanics equilibrium properties of the system are recovered by all the different methods, the Nosé-Hoover and the Langevin thermostats fail in reproducing the nonequilibrium behavior of such Hamiltonian.
We study a stochastic perturbation of the Nosé-Hoover equation (called the Nosé-Hoover equation under Brownian heating) and show that the dynamics converges at a geometric rate to the augmented Gibbs measure in a weighted total variation distance. The joint marginal distribution of the position and momentum of the particles in turn converges exponentially fast in a similar sense to the canonical Boltzmann-Gibbs distribution. The result applies to a general number of particles interacting through wide class of potential functions, including the usual polynomial type as well as the singular Lenn…
In response to W. G. Hoover's comment [arXiv:1204.0312v2] on our work [arXiv:1203.5968], we show explicitly that the divergence of the velocity field associated with the Nosé-Hoover equations is nonzero, implying that those equations are not volume preserving, and hence, as often stated in the literature, are not Hamiltonian. We further elucidate that the trajectories {q(t)} generated by the Nosé-Hoover equations are generally not identical to those generated by Dettmann's Hamiltonian. Dettmann's Hamiltonian produces the same trajectories as the Nosé-Hoover equations only on a specific energy …
Ergodicity of the systems with Nosé-Hoover thermostat are studied. The dynamics of the heatbath variables are investigated and they can be periodic when the system has quick oscillation. The periodic behaviour of them causes the system to lose its ergodicity. The kinetic-moments method is also studied, and the heatbath variables in this method are found to be chaotic. The chaotic behaviour makes the whole system ergodic.
In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are globally controlled by means of a single additional Nosé variable, and another where each demon is coupled to an independent Nosé-Hoover thermostat. Numerically stable and efficient measure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaotic properties of the different phase space flows are numerically illustrated through …
One-variable thermostats are studied as a generalization of the Nosé--Hoover method which is aimed to achieve Gibbs' canonical distribution with conserving the time-reversibility. A condition for equations of motion for the system with the thermostats is derived in the form of a partial differential equation. Solutions of this equation construct a family of thermostats including the Nosé--Hoover method as the minimal solution. It is shown that the one-variable thermostat coupled with the one-dimensional harmonic oscillator loses its ergodicity with large enough relaxation time. The present res…
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.
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 aims 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 equili…
Campisi, Zhan, Talkner, and Haenggi state, in promoting a new logarithmic computational thermostat [ arXiv 1203.5968 and 1204.4412 ], that (thermostated) Nose-Hoover mechanics is not Hamiltonian. First I point out that Dettmann clearly showed the Hamiltonian nature of Nose-Hoover mechanics. The trajectories {q(t)} generated by Dettmann's Hamiltonian are identical to those generated by Nose-Hoover mechanics. I also observe that when the (Hamiltonian) Campisi thermostat is applied to "nonequilibrium" heat transfer problems some very interesting, and somewhat paradoxical, phase portraits can resu…
While thermostated time evolutions stand on firm grounds and are widely used in classical molecular dynamics (MD) simulations, similar methods for quantum MD schemes are still lacking. In the special case of a quantum particle in a harmonic potential, it has been shown that the framework of coherent states permits to set up equations of motion for an isothermal quantum dynamics. In the present article, these results are generalized to indistinguishable quantum particles. We investigate the consequences of the (anti-)symmetry of the many-particle wavefunction which leads to quantum entangled di…
Employing a simple hydrophobic-polar heteropolymer model, we compare thermodynamic quantities obtained from Andersen and Nose-Hoover molecular dynamics as well as replica-exchange Monte Carlo methods. We find qualitative correspondence in the results, but serious quantitative differences using the Nose-Hoover chain thermostat. For analyzing the deviations, we study different parameterizations of the Nose-Hoover chain thermostat. Autocorrelations from molecular dynamics and Metropolis Monte Carlo runs are also investigated.
We present a derivation of the Jarzynski identity and the Crooks 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 non-equilibrium process and the Jacobian of the phase flow generated by the dynamics.
Nosé and Hoover's 1984 work showed that although Nosé and Nosé-Hoover dynamics were both consistent with Gibbs' canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs' Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs' canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of two-thermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tucke…
With the objective of demonstrating 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. Results from the molecular dynamics simulations in canonical ensemble, with various thermostats, are compared with those from 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, appears inadequat…
We present a general molecular-dynamics simulation scheme, based on the Nose' thermostat, for sampling according to arbitrary phase space distributions. We formulate numerical methods based on both Nose'-Hoover and Nose'-Poincare' thermostats for two specific classes of distributions; namely, those that are functions of the system Hamiltonian and those for which position and momentum are statistically independent. As an example, we propose a generalized variable temperature distribution that designed to accelerate sampling in molecular systems.
We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as iso-kinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless finite-time averages of the phase-space contraction rate have non-trivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for non-equilibrium stationary states, and appropriat…
We consider the models of relaxational dynamics within the framework of Berendsen's and Nose-Hoover's thermostats. On studying the crucial case of ideal gas we come to the conclusion that both models mentioned above do not allow for describing the true thermodynamical equilibrium.
A thermostat of the Nose-Hoover type, based on relative velocities and a local definition of the temperature, is presented. The thermostat is momentum-conserving and Galilean-invariant, which should make it suitable for use in Dissipative Particle Dynamics simulations, as well as nonequilibrium molecular dynamics simulations.
The model quantum system of fermions in a one dimensional harmonic oscillator potential is investigated by a molecular dynamics method at constant temperature. Although in quantum mechanics the equipartition theorem cannot be used like in the Nose-Hoover-thermostat it is possible to couple an additional degree of freedom to the system which acts as a thermometer and drives the system towards the desired temperature via complex time steps.
A fundamental challenge is to understand nonequilibrium statistical mechanics starting from microscopic chaos in the equations of motion of a many-particle system. In this review we summarize recent theoretical advances along these lines. Particularly, we are concerned with nonequilibrium situations created by external electric fields and by temperature or velocity gradients. These constraints pump energy into a system, hence there must be some thermal reservoir that prevents the system from heating up. About twenty years ago a deterministic and time-reversible modeling of such thermal reservo…
In the companion paper [Ingebrigtsen et al., arXiv:1012.3447] an algorithm was developed for tracing out a geodesic curve on the constant-potential-energy hypersurface. Here simulations of this NVU dynamics are compared to results for four other dynamics, both deterministic and stochastic. First, NVU dynamics is compared to the standard energy-conserving Newtonian NVE dynamics by simulations of the Kob-Andersen binary Lennard-Jones liquid, its WCA version (i.e., with cut-off's at the pair potential minima), and the Gaussian Lennard-Jones liquid. We find identical results for all quantities pro…
