314 The Monte Carlo method
This form of grand canonical Monte Carlo was presented by Norman and Filinov
[23]. Other approaches turn particles into ‘ghost’ particles instead of annihilating
them [ 24 , 25 ], and these ghosts can be switched on so that they enter the real life
of the simulation again. The Norman–Filinov version is, however, more popular.
It is possible to change the relative rates of the creation and annihilation process
and correct for this by a suitable change in acceptance rates of the corresponding
trial configurations. This was done for example by Saito and Müller-Krumbhaar
[26]; see also Problem 10.3. These methods may be useful in situations where the
acceptance rate for particle creation in the standard method becomes exceedingly
small because of a high value of the chemical potential.
It should be noticed that at high densities, insertion of new particles is likely to fail
because the probability of spatial overlap between the new particle and the existing
ones becomes very high. In this case, the Boltzmann factor is small, not as a result
of a high value of the chemical potential but as a result of the interactions between
the new particle and the existing ones. Methods have been devised for locating
‘cavities’ in such fluids and creating particles preferentially in these regions [27].
More details on grand canonical Monte Carlo methods can be found in
Refs. [ 2 , 28 ].
10.4.3 The Gibbs ensembleStudying the coexistence of different phases of the same material, and of different
species that can transform into each other via chemical reactions, is difficult using
the ensembles defined up to now. We know that for coexistence to be possible, the
different phases or species must have equal temperature, pressure and chemical
potential. We shall use the name ‘species’ in the context of both chemical mix-
tures and phase transitions in the following. If we fix pressure and temperature, the
chemical potentials of the different species will in general not be equal and one
species will grow at the expense of the other, until either one of the two has disap-
peared or until the chemical potentials are equal. However, it is often very hard to
achieve equilibrium in such cases since droplets surrounded by domain walls with
a nonvanishing wall tension need a long time to disappear.
Panagiotopoulos [29, 30 ] has developed a method in which two subsystems are
considered with no interface between them so that they are free to exchange particles
without having to overcome free energy barriers. The method is called ‘Gibbs
ensemble method’. The method is quite simple. Consider a volumeVwhich is
divided into two subsystems with volumesV 1 andV 2 by a freely movable piston. The
volumeV=V 1 +V 2 is fixed, so the total system is described by the(NVT)ensemble.
Furthermore, there is a virtual hole in the piston through which particles can move
from one subsystem to the other. Most importantly: there are no interactions between
the particles of the two subsystems; that is, if a particle moves fromV 1 toV 2 , the
energy difference it feels consists of the interactions with its partners in the new