Rescale Thermostat

One obvious way to control the temperature of a system is to rescale the velocities of the atoms within the system (Woodcock 1971), The rescaling factor A is determined from A /Ttarget/To, where Ttarget and To are the target and initial temperatures, respectively. Then, the velocity of each atom is rescaled such that Vf = AV . In practice, the inputs generally required to use a rescale thermostat include  

T - Damping constant (i.e, frequency with which to apply the thermostat) 

8T - Maximum allowable temperature difference from Ttarget before thermostat is applied /rescale Fraction of temperature difference between current temperature and Ttarget is corrected during each application of thermostat 

If it is desired to have a strict thermostat (i.e., when first starting a simulation that might have particles very near one another), then ST and r should have values of 0.01 Ttarget and 1 time step, respectively, and /escaie should be near 1.0. However, if you wish to allow a more lenient thermostat, then the value of ST should be of the same order of magnitude as Target, t should be 10 -10 time steps, and/escaie 0.01-0.1. 

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Obviously the amount of control that the thermostat imposes on the simulation is controlled by the value of r. If r is large, then the coupling will be weak and the temperature will fluctuate significantly during the course of the simulation. While if r is small, then the coupling will be strong and the thermal fluctuations will be small. If t = St, then the result will be the same as the rescale thermostat, in general. 

Another method of controlling the temperature that can be used in CP MD is the stochastic thermostat of Andersen.27 In this approach the velocity of randomly selected nucleus is rescaled this corresponds in a way to the stochastic collisions with other particles in the system. Therefore, this approach is often called a stochastic collision method. The Andersen thermostat has recently been shown28 to perform very well in the Car-Parinello molecular dynamic simulations of bimolecular chemical reactions. 

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. 

An alternative stochastic-dynamical method for thermostatting molecular dynamics is the stochastic velocity rescaling method proposed by Bussi and collaborators [60, 63] and somewhat generalized in [225]. The equations are (for the case Af = /, considered for simplicity) ... 

To understand the reason for this fundamental, qualitative difference between Nos6-Hoover-Langevin and Langevin dynamics, and to compare it with Velocity Rescaling, one may study a quantity that relates the rate of convergence to equilibrium to the rate of growth of the error in the autocorrelation function. In [225], this precise quantity is introduced and termed the efficiency of the thermostat ... 

Integrators and Thermostatting. - Control of temperature in a MD simulation is a desirable feature in many applications, particularly away from equilibrium. Woodcock was the first to devise a thermostatting procedure, by velocity rescaling. Since then, many thermostatting procedures have been devised. One of the most popular has been due to Berendsen, encapsulated in the following equations of motion. 

With this said we see that, when using any one of the thermostats described by Eqs. (80), (81), and (82), the time constant chosen should not be too small to allow the energy to redistribute equally between the different degrees of freedom after it is artificially pumped into the system or taken from it by velocity rescaling. 

Another way to control the temperature is to couple the system to an external heat bath, which is fixed at a desired temperature. This is referred to as a Berendsen thermostat (Berendsen et al. 1984). In this thermostat, the heat hath acts as a reservoir of thermal energy that supphes or removes temperature as necessary. The velocities are rescaled each time step, where the rate of change in temperature is proportional to the difference in the temperature in the system T t) and the temperature of the external hath Tbatn ... 

Thermostating is required in any non-equilibrium MFC simulation, where there is viscous heating. A basic requirement of any thermostat is that it does not violate local momentum conservation, smear out local flow profiles, or distort the velocity distribution too much. When there is homogeneous heating, the simplest way to maintain a constant temperature is to just rescale velocity components by a scale factor 5, = Sva, which adjusts the total kinetic energy to the desired value. 

This can be done with just a single global scale factor, or a local factor which is different in every cell. For a known macroscopic flow profile, u, like in shear flow, the relative velocities v - u can be rescaled. This is known as a profile-unbiased thermostat however, it has been shown to have deficiencies in molecular dynamics simulations [41],... 

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