Nose-Hoover equations of motion

The basic mechanism inherent in the Nose-Hoover equations of motion is that the variable p acts as a dynamic friction coefficient that controls the... 

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 proof that the Nose-Hoover thermostat samples a canonical ensemble of microstates provided that g = is as follows [63]. Consider the Nose-Hoover equations of motion, Eqs. (79) and Eq. (81). Because the variables r, p, and y are independent, the flow of the (2Ndf + l)-dimensional probabihty density p(r, p, y) is given by the generalized (non-Hamiltonian) analog of the Liouville equation ... 

Using these results and the definition of T, it is easily shown that dp/dt = 0 in Eq. (Ill) provided that g = Ndf. This shows that the extended-system phase-space density pe,r(i , p, y) is a stationary (equilibrium) solution of Eq. (111) corresponding to the Nose-Hoover equations of motion. Integrating out the y variable leads to the real-system phase-space probability density for the Nosd-Hoover thermostat (realtime sampling)... 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

Write and explain the equation of motion by Nose-Hoover. 

The most popular way to control the temperature in the CP MD simulation was introduced by Nose and Hoover.23,24 This approach includes an extra friction term (velocity dependent) into the Car-Parinello equations of motions (cf. Eq. 3) ... 

The time-step of 0.5 fs is used to simulate the dynamic system to 4.0 ps. The temperature of 300 K is used throughout the simulations. The MD simulations are performed using the Nose-Hoover thermostat for temperature control. The Hellmann-Feynman forces acting on the atoms are calculated from the ground-state electronic energies at each time step and are subsequently used in the integration of Newton s equation of motion. 

Constant temperatnre is maintained by Nose-Hoover thermostat and the equations of motion were integrated using the two time scale r-RESPA with a large time step of 2 fs and a small time step of 0.2 fs. Equilibration using these initial configurations was then carried out for at least 2 ns before beginning any produc-... 

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... 

Standard classical equations of motion that can be thought of as either scaling time or inertia, so that the system spends more time in the regions of phase space where the potential energy is a minimum. The equation of motion in Nose-Hoover form is... 

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... 

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. 

The first of these, proposed by Martyna, Tuckerman, and Klein (MTK), was based on the notion that the variable py, itself, has a canonical (Gaussian) distribution exp(- 3p /Q). However, there is nothing in the equations of motion to control its fluctuations. MTK proposed that the Nose-Hoover thermostat should, itself, be connected to a thermostat, and that this thermostat should also be connected to a thermostat. The result is that a chain of thermostats is introduced whereby each element of the chain controls the fluctuations of the element just preceding it. The equations of motion for such a thermostat chain are ... 

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - " are ... 

The theory that was developed in the previous section can be applied to the d mamics of a free Nose-Hoover particle with the associated equations of motion... 

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 ... 

Similarly, if thermostating (e.g. using a Nose-Hoover thermostat [169,170]) is performed while integrating the equations of motions for the nuclei, this will generate the canonical distribution... 

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)...

W. G. Hoover, Computational Statistical Mechanics, Elsevier, New York, 1991. [An alternative derivation of the canonical distribution is possible by incorporating temperature as a constraint in the equations of motion of the molecules. The resulting constrained equations are known as Nose-Hoover mechanics. For more details, see this reference and the references cited therein.]... 

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. 

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