Berendsen thermostat

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  

In O Eqs. 7.91 and O 7.92, r is the damping constant for the thermostat. In practice, the necessary inputs when using the Berendsen thermostat include  

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. 

---

Water molecules (density 1 g cm ) were placed uniformly within the periodic cell. Dynamics simulations were carried out at 298 K for 100 ps. Time steps were set at 0.001 ps and a Berendsen thermostat was used with a time step of 0.5 ps. Typically, 20 ps were used for equilibration runs, followed by production runs of 80 ps. 

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 initial state of the simulations consisted of RDX perfect crystals using simulation cells containing 8 molecules (one unit cell, 168 atoms) and 3D periodic conditions. After relaxing the atomic positions at each density with low temperature MD, we studied the time evolution of the system at the desired temperature with isothermal isochoric (NVT ensemble) MD simulations (using a Berendsen thermostat the relaxation time-scale associated with the coupling between the thermostat and the atomistic system was 200 femtoseconds). 

In this way, the kinetic energy of the system is scaled and the temperature of the system is forced to approach the prescribed temperature. The coupling time constant x represents the time scale in which the system reaches the prescribed temperature. Using the Berendsen thermostat, the system temperature is allowed to fluctuate instead of being set as a constant. At each time step, the temperature is corrected to a value closer to Tq. For a larger time constant (t), it takes longer time for a system to reach the prescribed temperature. 

Thermodynamic and dynamic properties were calculated over 700 000 steps after previous 200 000 equilibration steps. The time step was 0.001 in reduced imits, the time constant of the Berendsen thermostat (Berendsen et al., 1984) was 0.1 in reduced units. The internal bonds between the particles in each dimer remain fixed using the SHAKE (Ryckaert et al., 1977) algorithm, with a tolerance of 10 and maximum of 100 interactions for each bond. 

While the Berendsen thermostat is efficient for achieving a target temperature within your system, the use of a thermostat that represents a canonical ensemble once the system has reached a thermal equilibrium. The extended system method, which was originally introduced by Nosd [1984a, b] and then further developed by Hoover (1985), introduces additional degrees of freedom into the Hamiltonian that describes the system, from which equations of motion can be determined. 

Many of the approaches used for controlling the pressure are similar to those that are used for controlling the temperature. One approach is to maintain constant pressure by coupling the system to a constant pressure reservoir as is done in the Berendsen barostat (Berendsen et al. 1984), which is analogous to the way temperature is controlled in the Berendsen thermostat The pressure change in the system is determined by... 

A weaker formulation of this approach is the Berendsen thermostat where to keep temperature constant, system is coupled to an external heat bath of temperature To. The velocities are scaled in such a way that the rate of change of temperature is proportional to the difference in temperature between system and bath, that is. 

The Berendsen thermostat is quite efficient for relaxing a system to the target temperature. However, once yom system has reached equihbrium, it is important to probe a correct canonical ensemble. The Nose-Hoover thermostat is an extended-system method for controlling the temperatare of simulated system (Huenenberger 2005 Thijssen 1999). It allows temperatures to fluctuate about an average value, and uses a friction factor to control particle velocities. This particular thermostat can oscillate when a system is not in equilibrium. 

Therefore, it is recommended to use a weak-coupling method for initial system preparation (e.g., Berendsen thermostat), followed by data collection under the Nose-Hoover thermostat. This thermostat produces a correct kinetic ensemble. 

Inserting into Eq. (18) shows that the equation of motion corresponding to the Berendsen thermostat is... 

Assuming a relationship of the form of Eq. (115), it is possible to derive the configurational partition function of the weak-coupling ensemble as a function of a ([109] see Appendix) The limiting cases a = 0 (tb — 0 canonical) and a = 1 (tb —> oo microcanonical) are reproduced. Note that the Haile-Gupta thermostat generates configurations with the same probability distribution as the Berendsen thermostat with a = 1 /2. 

To obtain the corresponding Lagrangian equations of motion. Eg is initially treated as a constant and later expanded using Eq. (83). It appears that Eq. (78) has exactly the form of Eq. (18), i.e., the Nose-Hoover thermostat has one equation of motion in common with both the Woodcock/Hoover-Evans and the Berendsen thermostats. However, in contrast to these other thermostats where the value of y was uniquely determined by the instantaneous microstate of the system (compare Eq. (79) with Eqs. (45), (50), and (56)), y is here a dynamical variable which derivative (Eq. (79)) is determined by this instantaneous microstate. Accompanying the fluctuations of y, heat transfers occur between the system and a heat bath, which regulate the system temperature. Because y = s s = y = s (Eq. (77)), the variable y in the Nose-Hoover formulation plays the same role as s in the Nose formulation. When y (or s) is negative, heat flows from the heat bath into the system due to Eq. (78) (or Eq. (60)). When the system temperature increases above To, the time derivative of y (or s) becomes positive due to Eq. (79) (or Eq. (63)) and the heat flow is progressively reduced (feedback mechanism). Conversely, when y (or s) is positive, heat is removed from the system until the system temperature decreases below To and the heat transfer is slowed down. 

Here, the phase-space probability distributions are derived that correspond to the Woodcock/Hoover-Evans (Sect. 3.4), Haile-Gupta (Sect. 3.4), Berendsen (Sect. 3.5), Nosd (Sect. 3.6) and Nose-Hoover (Sect. 3.6) thermostats. The derivations are given for all but the Berendsen thermostat, for which the result is merely quoted. 

The derivation of the phase-space probability distribution for the Berendsen thermostat follows again similar lines [109], The final expression relies on the assumption of a relationship... 

---