Generation of the NPT Ensemble

The original work of Andersen and of Parrinello and Rahman on the generation of the NPT or isothermal-isobaric ensemble using an extended phase space predates Nose s work on the NVT ensemble, as noted above. Applying the extended system method to generate the NPT ensemble involves the inclusion of the volume into the phase space as a dynamical variable along 

Note that the last term in Eq. [77] involves a derivative of the potential with respect to the explicit volume dependence. Such an explicit dependence comes about when employing, for example, long range corrections to Lennard-Jones potentials or when Ewald summation is used to obtain energies and forces from electrostatic interactions. Equations [76] have the conserved energy 

For simplicity, Eqs. [76] show the coupling of the system to an ordinary Nose-Hoover thermostat. However, in practice, any desired thermostating scheme can be used. In addition, the system variables (p, q] as well as the volume are shown coupled to the same thermostat. In practice, it is useful to couple the system and the volume to separate thermostats, because the time scales of their fluctuations are usually considerably different. 

The phase space compressibility corresponding to Eqs. [76] can be expressed as follows  

Equations [76] only allow for isotropic fluctuations in the volume. However, it is sometimes useful to allow the lengths and angles of the simulation box all to vary separately. Fully flexible cell simulations of this type, first carried out by Parrinello and Rahman, can also be formulated in rms of a non-Hamiltonian dynamical scheme. In such a scheme, the matrix h representing the cell, which contains the cell vectors in its columns, is incorporated as a dynamical variable. That is, nine extra variables are added to the phase space along with an additional nine from its corresponding momentum Pg matrix. In terms of the box matrix, the partition function A(N, P, T) is given by 

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In the earlier subsection on the Dynamical Generation of the NPT Ensemble, we introduced equations of motion to perform equilibrium MD under constant temperature and pressure conditions. These equations of motion can be augmented with terms involving the shear rate from the SLLOD equations and can be written as follows ... 

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