Gaussian thermostat

In derivation of the steady state relations, artificial, reversible thermostats such as the Gaussian thermostat or Nose-Hoover thermostat have been used. However, Williams et al.22S have recently used MD simulations to verify assumptions made in their derivation that shows that the FR is insensitive to the details of the thermostatting mechanism. 

We emphasise that this is only a preliminary study of the stability properties of this system and questions related to the observed existence of different ergodic components in the phase space at particular values of the shear rate has not been characterized. One may wonder whether the complex dynamics observed at intermediate shear rates is linked with the use of the Gaussian thermostat. Extensive computations with other thermostats imply that the answer to this question is no , although actual orbits are, of course, affected by the particular thermostats. Some brief remarks on this point are made next. 

Instead of the Gaussian thermostat, which keeps the total kinetic energy of the system constant, the Nose-Hoover thermostat [19] is also frequently used in NEMD computer simulations [10], [14-17]. For this thermostat, the kinetic energy fluctuates around its average value determined by the prescribed temperature T, and in equilibrium, its dis-... 

SLLOD algorithm, the thermostat Gaussian multiplier j/ is introduced ... 

The thermostat affects the trajectories of the system. No real system evolves according to the Gaussian equations of motion. However, at equilibrium when the external field is equal to zero, ensemble averages of phase functions and time correlation functions are unaffected by the thermostat [14]. It is also possible to prove that the effects of the thermostat are qua(iratic in the external field and that the zero field limit of the linear response relation (2.17) is unaf-... 

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 original scheme utilized Gaussian isokinetic thermostats, whereas in Eqs. [217] we have replaced it with a Nose-Hoover thermostat. In this equation, the true local streaming velocity is given by iyy, + Usi(q,). In principle, there are no restrictions on u i, so the steady state velocity can be of any form hence, Evans and Morriss refer to Eqs. [217] as profile-unbiased thermostats (PUT). The PUT scheme requires only a reasonable prescription for determining the true local streaming velocity. 

Y. Liu and M. E. Tuckerman (2000) Generalized Gaussian moment thermostatting A new continuous dynamical approach to the canonical ensemble. J. Chem. Phys. 112, pp. 1685-1700... 

The continuous metadynamics algorithm can be applied to any system evolving under the action of a dynamics whose equilibrium distribution is canonical at an inverse temperature 1// . In a molecular dynamics scheme this requires that the evolution is carried out at constant temperature, by using a suitable thermostat [51]. In the continuous version of metad3mamics, Gaussians are added at every MD step and act directly on the microscopic variables. This generates at time t extra forces on x that can be written as... 

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. 

The starting point here is the Gaussian isokinetic thermostat [124], This type of constraint can be introduced into Newtonian dynamics q = M p, p = F q) by adding a Lagrange multiplier, i.e. changing the equations of motion to... 

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]. 

Figure 12. The reduced rotation J velocity, as inferred from the angular momentum (black) and from the components of the gyration tensor (gray) as functions of the shear rate for calculations with a Gaussian and an additional cxmi urational thermostat. The horizontal dashed line marks the limiting small shear rate v ne 0.5, the inclined dashed line shown for high shear rates corresponds to a power law exponent —0.75.

There are two different ways of accomplishing this. The first is to introduce some random perturbations to the system, which emulate its interaction with the enviromnent. For constant temperature, for instance, this can be done by randomly [50] changing momenta by some random increments drawn from the Gaussian distribution. This mimics collisions between the molecules in the system and the virtual thermostat particles. The choice of parameters for the Gaussian distribution of momentum increments is determined by the required temperature. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

Another popular thermostat used in molecular dynamics simulations is the Langevin thermostat. It covers the heat-bath coupling part of the Langevin equation by friction and Gaussian random forces f. The Langevin equation basically describes the dynamics of a Brownian particle in solvent under the influence of external forces F. Its simplest form therefore reads ... 

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