Nose-Hoover Chains

Natural imits were used in these simulations, where m = = ku = 1. The sampling of the initial position and momentum centroids were made through the Nose-Hoover chain dynamics (NHC) on the effective potential of Vcm, More details of these calculations can again be foimd in Ref 10. 

Liquids and proteins are complex systems for which the smdy of dynamical systems has wide applicability. In the conference, relaxation in liquids (s-entropy by Douglas at the National Institute of Standards and Technology, nonlinear optics by Saito, and energy bottlenecks by Shudo and Saito), energy redistribution in proteins (Leitner and Straub et al.), structural changes in proteins (Kidera at Yokohama City University), and a new formulation of the Nose-Hoover chain (Ezra at Cornell University) were discussed. Kidera s talk discussed time series analyses in molecular dynamics, and it is closely related to the problem of data mining. In the second part of the volume, we collect the contributions by Leitner and by Straub s group, and the one by Shudo and Saito in the third part. 

Equations [73], known as the Nose-Hoover chain (NHC) or MTK equations, have the conserved energy ... 

Although we have assumed in Eq. [209] that the velocity profile in the confined fluid is linear, it is not immediately obvious that this is technically possible in the absence of moving boundary conditions. A parallel to this situation is the comparison between Nose-Hoover (NH) thermostats and Nose-Hoover chain (NHC) thermostats. Although the Nose-Hoover equations of motion can be shown to generate the canonical phase space distribution function, for a pedagogical problem like the simple harmonic oscillator (SHO), the trajectory obtained from the NH equations of motion has been found not to fill up the phase space, whereas the NHC ones do. The SHO is a stiff system and thus to make it ergodic, one needs additional degrees of freedom in the form of an NHC.2 ... 

The Nose-Hoover chains method is expressed in terms of the non-Hamiltonian dynamical system with the following equations of motion ... 

Let us now examine the case of a one-dimensional free particle coupled to a Nose-Hoover chain of length M = 2 (which is analogous to the system that was already examined using the NosAHoover method). This system is defined in terms of the following equations of motion ... 

Figure 2 shows the results for simulations of the same one-dimensional harmonic oscillator with Nose-Hoover chains of var3ung lengths. The first column shows the results for a chain of length M = 1, which is equivalent to the NosAHoover algorithm. Notice that neither the momentum (b) nor position (c) distributions are canonical in nature. This is further confirmed by the presence of a Hoover hole in the Poincare section depicted in (a). 

Conversely, the oscillators that have been coupled to Nose-Hoover chain thermostats with length M = 3,4 result in momentum and position distributions that match the canonical ensemble distributions. Additionally, the Hoover hole has been eliminated from the Poincare sections for these cases. 

Fig. 2. Simulations of a one-dimensional harmonic oscillator coupled to Nose-Hoover chains of length M = 1 (a-c), M = 3 (d-f), and M = 4 (g-i). (a),(d),(g) The Poincare sections for these oscillators. (b),(e),(h) The momentum distribution functions. (c),(f),(i) The position distribution functions...

G. Martyna, M. Tuckerman, and M. Klein (1992) Nose-Hoover chains - the canonical ensemble via continuous dynamics. J. Chem. Phys. 97, p. 2635... 

In contrast, a system in contact with a thermal bath (constant-temperature, constant-volume ensemble) can be in a state of all energies, from zero to arbitrary large energies however, the state probability is different. The distribution of the probabilities is obtained under the assumption that the system plus the bath constimte a closed system. The imposed temperature varies linearly from start-temp to end-temp. The main techniques used to keep the system at a given temperature are velocity rescaling. Nose, and Nos Hoover-based thermostats. In general, the Nose-Hoover-based thermostat is known to perform better than other temperature control schemes and produces accurate canonical distributions. The Nose-Hoover chain thermostat has been found to perform better than the single thermostat, since the former provides a more flexible and broader frequency domain for the thermostat [29]. The canonical ensemble is the appropriate choice when conformational searches of molecules are carried out in vacuum without periodic boundary conditions. 

S. Jang and G. A. Voth, /. Chem. Phys., 107,9514 (1997). Simple Reversible Molecular Dynamics Algorithms for Nose-Hoover Chain Dynamics. 

Numerical methods for Nose-Hoover chains are easily constructed by splitting of the equations of motion. 

Fig. 8.1 Trajectories of the Nose-Hoover Chain system may be trapped forever in restricted regions of phase space. Here a toms is shown in a projection of the full extended phase space for the harmonic oscillator with a length two thermostatting chain. (Equations of motion q = p p = -q-hpili = (p - 1)// 1 -hhih = 1)/M2. with/ti = 0.2, and/t2 = 1)...

Jang, S., Voth, G. Simple reversible molecular dynamics algorithms for Nose-Hoover chain dynamics. J. Chem. Phys. 107(22), 9514-9526 (1997). doi 10.1063/1.475247... 

In order to obtain a steady state from Eqs. 38 dissipative heat must be removed from the system. This is achieved by the last (thermostatting) terms of the last two equations in Eqs. 38. In this respect it is essential to observe that accurate values for Uj and A are needed. Any deviations from the assumed streaming and angular velocity profiles (biased profiles) will exert unphysical forces and torques which in turn will affect the shear-induced translational and rotational ordering in the system [209,211,212]. The values for the multipliers and depend on the particular choice of the thermostat. A common choice, also adopted in the work of McWhirter and Patey, is a Gaussian isokinetic thermostat [209] which insures that the kinetic and rotational energies (calculated from the thermal velocities p" and thermal angular velocities ot) - A ) and therefore the temperature are conserved. Other possible choices are the Hoover-Nose or Nose-Hoover-chain thermostats [213-216]. 

To obtain the average values of properties of the particles at a fixed temperature and examine dependence of the conformations on temperature, we used Nose-Hoover chain (NHC) constant temperature molecular dynamics [228]. The initial configmations of the steady state of the amorphous PE particle were used at a start of the NHC simulations the initial values of the Cartesian momenta were given random orientation in phase space with magnitudes chosen so that the total kinetic energy was the equipartition theorem expectation value [229]. The NHC quasi-Hamiltonian for this system can be written,... 

For the determination of equilibrium properties, NVT (constant number of particles, volume and temperature) and NPT (constant number of particles, pressure and temperature) ensembles were used. For these ensembles, the use of a non-Hamiltonian system is required. Temperature was controlled with a thermostat based on a Nose-Hoover chain [17]. In each case, calculations were performed with a time step fixed at 1.2 fs. A rigid-ion model based on the Bom-Mayer potential (Equation 3.4.1) was used. This model takes into account the Van der Waals and Coulomb parameters. 

A. Cheng and K.M. Merz Jr., Application of the Nose-Hoover chain algorithm to the study of protein dynamics, J. Phys. Chem., 100 (1996) 1927. 

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