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Other statistical ensembles

  We have discussed so far the standard molecular dynamics scheme, based on the time integration of Newton's equations and leading to the conservation of the total energy. In the statistical mechanics parlance, these simulations are performed in the microcanonical ensemble, or NVE ensemble: the number of particles, the volume and the energy are constant quantities.

The microcanonical average of a physical quantity A is obtained as a time average on the trajectory:  
 \begin{displaymath}
\langle A\rangle_{\rm NVE} = \frac{1}{N_T}
\sum_{t=1}^{N_T} A(\Gamma(t))\end{displaymath} (26)
where I denote with $\Gamma(t)$ the phase space coordinate of the system (3N positions and 3N velocities).

There are other important alternatives to the NVE ensemble, that we will mention here only briefly. The basic idea is that of integrating other equations in place of Newton's equations, in such a way that sampling is performed in another statistical ensemble. Averages of physical quantities in the new ensemble will be again obtained as time averages, similarly to eq. (3.18).

A scheme for simulations in the isoenthalpic-isobaric ensemble (NPH) has been developed by Andersen [22]. Here, an additional degree of freedom representing the volume of the box has been introduced, and all the particle coordinates are given in units relative to the box. The volume V of the box becomes a dynamical variable, with a kinetic energy and a potential energy which just PV, where P is the external pressure. The enthalpy H=E+PV is a conserved quantity.

Parrinello and Rahman [23] developed a variant where the shape of the box can vary as well as the volume. This is achieved by introducing 9 new degrees of freedom instead of 1: the components of the three vectors spanning the MD box. Each of them is a new dynamical variable, evolving accordingly to equation of motion derived from an appropriate Lagrangian. This scheme allows to study structural phase transitions as a function of pressure, where for example the system abandons a certain crystal structure in favor of a more compact one.

Another very important ensemble is the canonical ensemble (NVT). In a method developed by Nosè and Hoover [24], this is achieved by introducing a time-dependent frictional term, whose time evolution is driven by the imbalance between the instantaneous kinetic energy and the average kinetic energy (3N/2) kB T.


next up previous contents
Next: Interatomic potentials Up: Running, measuring, analyzing Previous: Annealing and quenching: MD
Furio Ercolessi
9/10/1997