310 The Monte Carlo method
programming exercise
Code the Metropolis method for argon, and compare the results with those of
the molecular dynamics program; seeTable 8.1.
The heat-bath method turns out impractical in this example. If we were to use
the pure heat-bath method, we would have to allow the particle to move to any
position in the system cell, and then accept or reject this position using the Von
Neumann algorithm (seeAppendix B3), but as the vast majority of positions in the
cell are unacceptable, this is very inefficient. One could imagine a ‘hybrid’ heat-
bath method, in which we move the particle to a new position in a small sphere or
cube centred at the old position with a probability distribution determined by the
conditional HamiltonianH(ri|R−ri), using the symbolic notation ofEq. (10.22).
This can again be done using the Von Neumann method (seeAppendix B3). The
difference with the Metropolis method is that the new position is accepted with a
probability which is independent of the previous one (except for the sphere or cube
being centred on the old position). To apply the Von Neumann method, we should
know the minimum of the conditional Hamiltonian, as the acceptance probability in
the Von Neumann method may never exceed 1. We might guess a lower bound for
this minimum, but this will often be much lower than the actual potential minimum,
so that the acceptance rate becomes very small. Because of these difficulties, the
heat-bath method is not used for atomic or molecular systems.
Just as in the case of the Ising model, we may sometimes suffer from critical slow-
ing down due to a diverging the correlation time close to a critical point. Methods
have been developed which dramatically reduce this effect. A recent breakthrough
in this field is the Liu–Luijten algorithm[16].
10.4 Other ensemblesThe canonical ensemble is the most natural ensemble for MC simulations. It is,
however, possible to simulate other ensembles by the Metropolis MC method. We
shall consider the isothermal-isobaric, or(NPT)ensemble, and the grand canonical
ensemble. There also exists a microcanonical MC method,[17]but this is seldom
used as it is of little practical importance.
10.4.1 The(NPT)ensembleThe(NPT)ensemble is relevant because temperature and pressure are often kept
fixed in experiments. A Monte Carlo method for this ensemble was first developed
for hard sphere systems by Wood [ 18 , 19 ] and later extended to smooth potentials
by McDonald [ 20 , 21 ]. We consider the latter case here.