Nose-Hoover chain thermostat

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. 

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. 

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]. 

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 ... 

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)...

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. 

For the finite-temperature simulations, the temperature of the Si ions were controlled with a chain of five, linked Nose-Hoover thermostats." Because the electrons are always quenched back onto the Born-Oppenheimer surface after every timestep, no additional thermostat is needed for the electrons. Details of the configurations were similar to those with the CP scheme, except that the in-plane cells consisted of 16 atoms per layer and the basic timestep of the simulation was 100 a.u. 

The Nose-Hoover thermostat exhibits non-ergodicity problems for some systems, e.g. the classical harmonic oscillator. These problems can be solved by using a chain... 

The Nose-Hoover thermostat, or chain of thermostats, can be used as well to control the wave function temperature, i.e. the fictitious kinetic energy. This prevents drifting of the wave function from the Born-Oppenheimer PES during long simulations. Wave function thermostats are introduced in a similar way to Eqs. 7-9. 

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 ... 

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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