Nose-Hoover methods

In Nose-Hoover methods the heat bath is considered an integral part of the system, and enters the simulation on an equal footing with the other variables. 

Minimum Energy Path (MEP), 344, 390 Nose-Hoover method, tor sunulations, 386 solvahon, 396 ... 

The methods described above address the solution to Newton s equations of motion in the microcanonical NVE) ensemble. In practice, there is usually the need to perform MD simulations under specified conditions of temperature and/or pressure. Thus, in the literature there exist a variety of methodologies for performing MD simulations under isochoric or isothermal conditions [2,3]. Most of these constitute a reformulation of the Lagrangian equations of motion to include the constraints of constant T and/or P. The most widely used among them is the Nose-Hoover method. 

After energy minimization, the simulations were run at constant pressure using a Langevin piston Nose-Hoover method as implemented in the NAMD software package until the system had reached its equilibrium volume at a pressure of 1 atm and 300 K in the NPT (constant particle number, pressure, and temperature) ensemble. The system s behavior was then simulated for 200 ns (100 million steps) in the A F T (constant particle number, volume, and... 

Controlling the temperature implies specifying the parameters characteristic for the thermostat method used in the simulation. In the case of the most popular Nose-Hoover thermostat the basic quantities are the target temperature and the thermostat frequency. 

S. Nose, An Extension of the Canonical Ensemble Molecular Dynamics Method, Mol. Phys. 57 (1986) 187-191 S. Nose, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, J. Chem. Phys. 81 (1984) 511-519 D. J. Evans and B. L. Holian, The Nose-Hoover Thermostat, J. Chem. Phys. 83 (1985) 4069-4074 B. L. Holian, A. F. Voter and R. Ravelo, Thermostatted Molecular Dynamics How to avoid the Toda Demon Hidden in Nose-Hoover Dynamics, Phys. Rev. E 52 (1995), 2338-2347 Luis F. Rull, J.J. Morales and F. Cuadros, Isothermal Molecular-Dynamics Calculations, Phys. Rev. B 32 (1985) 6050-6052. 

Two of more sophisticated and commonly used approaches are the Nose-Hoover thermostat [79,80] and Langevin method [81]. In the Langevin method, additional terms are added to the equations of motion corresponding to a fiiction term and a random force. The Langevin equation of motion is given by... 

The Berendsen [82] and Gauss [83] thermostats are also among other methods used. The Berendsen thermostat [82] was developed starting from the Langevin formalism by eliminating the random forces and replacing the friction term with one that depends on the ratio of the desired temperature to current kinetic temperature of the system. The resulting equation of motion takes the same form as the Nose-Hoover equation with... 

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 ... 

The motion equations have been solved by the Verlet Leap-frog algorithm subject to periodic boundary conditions in a cubic simulation cell and a time step of 2 fs. The simulations have been performed in the NVT ensemble with the Nose-Hoover thermostat [62]. The SHAKE constraints scheme [65] was used. The spherical cutoff radius comprises 1.2 nm. The Ewald sum method was used to treat long-range electrostatic interactions. 

Here the constant k is the compressibility of the system. Such barostat methods are again widely used, both in MC and MD simulations, but do not produce strictly correct ensembles. Alternatively, the pressure may be maintained by a Nose-Hoover approach in order to produce a correct ensemble. 

As we have seen in earlier chapters, there is a strong foundation for using symplectic integrators, for example the existence of perturbative backward error expansions which are relatively easily calculated, it is desirable to work within this class. On the other hand, as we have mentioned, Nose-Hoover dynamics is not a Hamiltonian system. The Nos6-Poincare method resolves this issue by providing a Hamiltonian-based extension of the physical system whose projected trajectories are actually equivalent to Nos6-Hoover trajectories, in the absence of numerical errors. This extended Hamiltonian system can then be discretized using a symplectic method. 

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

It should also be mentioned that the measure-preserving thermostatting methods discussed in [373] refer rather to the underlying measure of the phase space and not the thermodynamic equilibrium distribution those methods are deterministic and, like Nose-Hoover based schemes, will exhibit thermodynamic errors due to both lack of ergodicity and discretization error, in addition to sampling errors. 

The Nose-Hoover-Langevin (NHL) method is based on a simple idea replace the chain in the Nos6-Hoover Chain, whose sole purpose is to maintain a Gaussian distribution in the auxiliary variable, by a stochastic Langevin-type thermostat. The method was first proposed in [323]. The proof of ergodicity (more precisely the confirmation of the Hormander condition), for a problem with harmonic internal interactions, was given in [226] and we roughly follow the treatment from this paper. 

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