# 6.1 NPT Monte Carlo

Samples the phase space of a constant-P, constant-T ensemble with the appropriate phase space probability. In addition to the particle moves performed at constant , changes of the volume are attempted and accepted/rejected according to an evaluation of the enthalpy change. A simple LJ simulation of the NPT type is sketched here:

For given and , let the instantaneous volume be and the particle positions , .
• Choose a particle and perform a trial move in the usual manner. Compute the energy difference between the trial configuration and the given one: , where etc.

Accept or reject the trial configuration with probability
 (6.1)

• Repeat this basic MC step for a number of particles; usually all particles are treated in sequence.
• Now perform a trial volume change ; all particle coordinates are (implicitely) scaled by , which entails a change in energy of .

Compute the enthalpy change
 (6.2)

and accept/reject the new volume according to
 (6.3)

• This ends the MC cycle.
As usual, the maximum particle step and the maximum volume change are adjusted to achieve an acceptance ratio near .

Note 1: If you are lucky, the model pair potential may be written in the scalable form where is a scaled distance, and is the scaling factor. In such cases the total potential energy after a volume change need not be recalculated from scratch; rather, we have . As an example, take the term in the Lennard-Jones potential. When scaling all distances from to , where , we have
 (6.4)

Note 2: A sample program for NPT MC (and for many other simulation techniques), may be found on the web page of Allen and Tildesley's textbook, www.ccl.net/cca/software/SOURCES/FORTRAN/allen-tildesley-book

Note 3: A JAVA applet for NPT MC (and for many other simulation techniques), may be found on the web page of David Kofke, U of Buffalo, www.cheme.buffalo.edu/courses/ce530/Applets/applets.html
F. J. Vesely / University of Vienna