316 The Monte Carlo method
It is not necessary to separate the two subvolumes by a movable piston in order
to arrive at equal pressures in both subsystems: it is also possible [30] to couple
both subsystems to a ‘pressure bath’ with predefined pressurePsimilar to the
(NPT) method described inSection 10.4.1, and imposing no restriction on the
total volumeV 1 +V 2. This method is less suitable for phase coexistence as the
coexistence occurs on a line in the (P,T) diagram. Therefore we need to know
the exact location of that line, as we have to specifyTandPin this method.
In the original version, the system will move to the coexistence line by adjusting
pressure and chemical potential simultaneously. In the case of chemical equilibrium,
however, the coexistence region has a finite width and the constant (P,T) version
is useful there. The Gibbs ensemble method has become very popular for studying
coexistence equilibria in recent years [ 29 – 32 ].
*10.5 Estimation of free energy and chemical potentialInSection 7.1we have already mentioned the difficulties involved in calculating
free energies. We described briefly the method of thermodynamic integration. Other
methods have been proposed and these are easier to understand in the context of MC
trials and this is the reason why we discuss them in this chapter. It should, however,
be noted that the methods described below are not restricted to MC. Some are
applicable within MD, especially in the canonical(NVT)MD method. In view of
the equivalence of ensembles, microcanonical MD allows for using these methods
too[33].
ThereaderisreferredtoRefs.[ 28 , 32 ]formoredetailsconcerningthematerial
in this section.
10.5.1 Free energy calculationThe difficulty in free energy calculation is that it cannot be formulated directly as a
‘mechanical average’, that is, an ensemble average of functions of the coordinates
ri(andpiin the case of MD). Rather, the free energy must be evaluated as a sum
(or integral) over phase space. Clearly, both MD and MC methods sample that part
of phase space from which the dominant contribution to the free energy arises;
however, this does not provide an estimate of the phase space volume integral,
as the frequency with which phase space points are visited is proportional to the
Boltzmann weight, with an unknown proportionality factor. Moreover, it is ques-
tionable whether the number of points visited in a typical simulation would be
sufficient to sample the relevant phase space volume adequately.
It is nevertheless possible to formulate the free energydifferencebetween two
systems with different interactions as a mechanical ensemble average. The first