Temperature and pressure controls

Here we discuss elaborations of the basic MD method ( see formula page) which allow a more direct comparison with experiment.

The basic MD ensemble is constant energy and constant volume. However, experimentally it is usually more convenient to specify the pressure and temperature. In those variables the phase of the system is unique except precisely at a phase transition. If we input the number of particles and volume near a phase boundary, then it is possible we will be in the two phase region, for example we could have a mixture of ice and water. This is unlikely to occur in a simulation because our system is usually so small. The interface between the two phases would cost too much energy and inhibits the system from moving freely from one phase to the other. Temperature controls and pressure controls allow one to do such simulations more directly.

Temperature Controls

We want to do simulations at constant temperature. How do we do it?

• Velocity rescaling. The simplest method is to start the simulation with random velocities. If the system is ergodic, the only thing that matters is the total energy. One could give all the extra energy to a single particle and the energy would cascade down to all the particles, and the velocity would have a normal distribution after a few collisions. But to minimize computer time, you want to start as close as possible to the final true distribution, a normal distribution. (Soon we will learn how to sample from a normal distribution.) In the velocity rescaling method, every so often one re-scales the velocity, multiplying by a constant chosen so that the kinetic energy is 3/2K T per particle. Once the system reaches equilibrium, the scaling is no longer necessary.
• Heat Bath. A more physically based approach is to put the particles in contact with a heat bath. We will discuss this in more detail later in the course. Algorithmically we imagine the particles colliding with fictitious particles. Instead of F=ma we also have random forces coming from these collisions and a friction term.
• Nose-Hoover dynamics. (Also known as a Nose thermostat.) An approach pioneered in 1984 by Nose adds a single heat bath particle. A new variable with the dimensions of a friction constant is added to the system. This approach gives reversible deterministic dynamics and if the system is ergodic we will be guaranteed to go to the canonical distribution. The approach of adding a new fictitious particle is very common trick in simulation.

The disadvantage of all the above methods for working at constant temperature is that the dynamics is changed in an artificial way and dynamics is the reason why we do MD and not MC. Real dynamics obeys the rules of special relativity. Effects need to be mediated by the transport of particles since we don't have any photons. Heat is transported at the speed of sound or slower. But if there is a kinetic energy fluctuation, the effect on the other particles with the various temperature thermostats is immediate. All three methods have a coupling constant (respectively the frequency of rescaling, the friction coefficient or the mass of the N-H variable.) It should be chosen as small possible to alter the dynamics as little as possible. If this is done, the non-physical effects will be order (1/N) in general. If this is a problem one should use a more physical method to control temperature fluctuations, for example put in a real heat bath at the edge of the physical system.
  A & T pgs. 227-231

Pressure Controls

Pressure controls can be introduced in a similar fashion. The conjugate variable to the pressure is the size of the box. Anderson, Parrinello and Rahman (1980-84) introduced a formalism where the size of the box is a dynamical variable. When the box size fluctuates (because the pressure from the virial is not equal to the desired pressure) all the particle positions dilate. See the text book for how this works out in practice. In the Parrinello-Rahman method, the box shape also fluctuates; it is allowed to become an arbitrary parallelpiped. Then the system can switch between different crystal structures by itself (for example between FCC and BCC). This method is very useful is studying the transitions between two different crystal phases.

Again the dynamics is unrealistic. In addition the size effects can be larger than in a cubic box because fluctuations in the size make the box narrower in some directions. Remember that just because a system can fluctuate from one structure to another does not mean than the probability is high for that to happen.

A & T pgs. 232-239

See formula page.

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© D. M. Ceperley 2000, D.D. Johnson 2001