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Calculation of temperature
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- Calculation of pressure
- 4. Environmental coupling: Thermostat, Barostat
- 4.1. Temperature control
Calculation of temperature As we have learned from statistical physics the temperature of molecular system containing N particles can be calculated from the average kinetic energy and the equipartition theorem .
Here kB is the Boltzmann constant, T is the absolute temperature.
Here viμ is the velocity of the i-th atom at the μ-th trajectory point of the simulation. Further M is the number of sampling points along the trajectory. The average kinetic energy was calculated using the above described expression (eq. 6.16) for calculation of averages of physical quantities. The actual temperature of our system at a time t during the simulation is
Calculation of pressure The calculation of pressure is a more complicated task. First, we have to introduce the concept of virial originated from Clausius [4]. The virial function (W) of a molecule (system built from mass points) is defined as
where Fitot is the total force acting on the i-th atom at position ri. It is easy to show that the time average of it is related to average of the kinetic energy as
Using the formula (eq K-average) above we obtain
As a next step we separate the Fi to internal (interaction of the atoms) and external (interaction with the environment) forces
Assuming that the external forces are originated from the interaction of the atoms with a cuboid-like container, the contribution of the external forces to the time average of the virial is
Fiext(τ) is non-zero only for those particles which bounce to the wall of the container. For the sake of simplicity let us suppose that our container has a form of a rectangular prism with edges ax,ay,az. One of its corners is located at the origin of a coordinate system its edges are parallel to the axes. The forces at the six walls are perpendicular to the wall and directed to the inside of the container. Using this, the non-zero contributions to the virial average are
The three terms can be rewritten using the definition of the pressure p,
where ayaz is the surface area of that side of the rectangular prism which is parallel to the y-z coordinate plane. With this
here V=ax ayaz is the volume of the container. Collecting our result
Or after rearrangement
It is worth to mention that this equation reduces to the equation of state of the perfect gas if there is no interaction between the particles. 4. Environmental coupling: Thermostat, Barostat To simulate thermodynamical ensembles we have to fix variables. Technically easy to tackle the problem if the volume (an extensive variable) is kept fixed in our simulation. We need simply to fix the geometrical parameters of the simulation box. However,it is more complicated to fix an intensive thermodynamical variable. As in reality, we need a thermostat or a barostat if we want to specify the temperature or the pressure of our system. The proper mathematical/computational representation is very important in MD because it inherently influences the probability distribution of the micro states obtained from our simulation and this way the calculated averages of thermodynamic quantities. Following the categorization of G. Sutmann [5] we can distinguish four different methods for controlling a thermodynamic quantity A: Differential control: The value of A is fixed, no fluctuations are allowed Proportional control: The variables influencing the actual value of A are corrected towards the prescribed value of A in each simulation step. The ‘speed’ of correction is determined by a coupling constant which also determined the fluctuation of A around its average value. This method simulates directly a system immersed in a ‘bath’ (e.g. thermostat or barostat). Integral control: In this case the environmental coupling is mimicked by adding extra degrees of freedom to the system which guaranties the prescribed value of A. Stochastic control: Certain degrees of freedom are modified stochastically to improve the control. 4.1. Temperature control Temperature control As we learned from statistical physics the temperature of molecular system containing N particles can be calculated from the average kinetic energy and the equipartition theorem: .
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