Nose-Hoover coupling

An alternative coupling scheme for temperature and pressure, the Nose-Hoover scheme, adds new, independent variables that control these quantities to the simulation (Nose 1984 Hoover 1985). These variables are then propagated along with the position and momentum variables. 

It was noted by Hoover that Eqs. [65] are not guaranteed to yield ergodic trajectories, in which case a dynamical simulation based on these equations of motion would not generate a canonical distribution in H p, q). This was seen most dramatically in the example of a single harmonic oscillator coupled to a Nose-Hoover thermostat, where a distribution radically different from the correct canonical distribution was generated as a result of nonergodicity. Thus far, two different solutions to this problem based on continuous dynamics have been proposed. 

For simplicity, Eqs. [76] show the coupling of the system to an ordinary Nose-Hoover thermostat. However, in practice, any desired thermostating scheme can be used. In addition, the system variables (p, q] as well as the volume are shown coupled to the same thermostat. In practice, it is useful to couple the system and the volume to separate thermostats, because the time scales of their fluctuations are usually considerably different. 

Once again, Eqs. [84] utilize only a single Nose-Hoover thermostat, although in practice, one should couple separate thermostats to the cell matrix and to the system variables. The ensemble average of p is, from Eqs. [84],... 

Here, we have coupled the GSLLOD equations to a Nose-Hoover thermostat. It is simple to show that the computation of q, in the unthermostated form of the GSLLOD equations will yield Eq. [133]. 

From the earlier section on Molecular Dynamics and Equilibrium Statistical Mechanics, we hope that we have made clear that a conserved quantity is the starting point for phase space analysis and the derivation of a probability distribution function. Following the same analysis that led to the distribution functions for NVE, NVT, and NPT dynamics, the new distribution /(q, p,, I) for GSLLOD coupled to a Nose-Hoover thermostat is given by... 

For SLLOD dynamics coupled to a Nose-Hoover thermostat, the metric determinant is given by The first and third terms in Eq. [229] become Nfe f feqPr /Q with Opposite signs and thus cancel. We are left with the expression... 

Fig. 1. The momentum distribution function for a one-dimension free particle coupled to a Nose-Hoover thermostat solid line) [rn = 1, kT = 1, p(0) = 1, q 0) = 0, Q = 1, Pr 0) = 1, time step At = 0.05] compared to the analytical result dashed line) and the canonical ensemble distribution dot dashed line)...

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 ... 

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...

In the case of Nose-Hoover, this problem was known to both of the inventors and has been observed by many authors (see e.g. [251] and [180]). More recently, it has been studied analytically [218], Some confusion arises from the fact that the practical ergodic properties (and other properties) in Nosd-Hoover simulations are sensitive to the particular system under study and the values of parameters. For example it is known that in the case of simple liquids and gases, Nos6-Hoover does (at least under some conditions) enable reliable calculation of thermodynamical quantities [77], whereas in other systems results are poor and cannot be rectified by adjustment of parameters. In general, the control of temperature requires that the thermostat couples tightly to the physical variables, which generally calls for a small thermal mass, but this may be at odds with efficiency reduction (it may lead to a need to use smaller timesteps to resolve the motion of the auxiliary variable). 

Nose-Hoover-Langevin An Ergodic Weakly Coupled Thermostat... 

Given a constant temperature and pressure MD code, it is not very difficult to modify it for GEMD. Experience also shows that the same results are obtained both with the Nose-Hoover constant temperature MD algorithm, which exactly reproduces the NVT) ensemble [2,3], and with the approximate but numerically more stable weak coupling approach [9]. 

Similar to the Nosd-Hoover thermostat, the extended system method has been applied to create a barostat (Hoover 1986) that is coupled with a Nose-Hoover thermostat. In this case, the extra... 

Therefore, it is recommended to use a weak-coupling method for initial system preparation (e.g., Berendsen thermostat), followed by data collection under the Nose-Hoover thermostat. This thermostat produces a correct kinetic ensemble. 

For the majority of atomic and small molecule systems at equilibrium in the (N,P,T) ensemble (P is the pressure tensor) it is widely accepted that the most rigorous approach is to use the controlled pressure technique proposed by Rahman and Parrinello (RP) in conjunction with the Nose-Hoover thermostat.However, the choice of method must take careful account of the material we wish to study, how it is modeled and any external perturbations which we wish to apply. For polymers the Berendsen loose-coupling controlled pressure MD technique is a good compromise. Although the theoretical basis of this method has been criticised in practice it has been found that to within statistical uncertainties first-order properties are the same as those obtained by more rigorous approaches. 

For the canonical dynamics simulation, the temperature (T) is held constant by coupling to a thermal bath. Nose (61) and Hoover (62) suggested different methods of thermal coupling for canonical dynamics. Canonical dynamics using Hoover s heat bath gives the trajectory of particles in real time, while the molecular dynamics based on Nose s bath does not give the trajectory of particles in real time due to its time scaling method. Therefore, for a real-time evaluation of the system, Hoover s heat bath should be used for the canonical MD. 

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