Computational Physics

10.6 Further applications and Monte Carlo methods 319

[36]. The method works well, although problems arise for high densities. In that
case the Boltzmann factor is very small for most trials, because the probability that
the core of the new particle overlaps with one of the existing particles becomes
very high. There have been refinements in which the particle insertions are biased
towards the cavities in the fluid [ 22 , 37 , 38 ].
Instead of particle insertions, particle removals could be used to find the chemical
potential. In this case we would use the inverse of Eq. (10.47):

ZN− 1

ZN

=

N!^3 N

(N− 1 )!^3 (N−^1 )

∫

dRN− 1 exp[−βU(RN− 1 )]

∫

dRNexp[−βU(RN)]

=NV^3 〈

1

V

∫

d^3 rN exp(βU−)〉N=e+βμ. (10.48)

However, generalisation of the method proposed above to this case usually fails as
the sampling of exp(βU−)is very inefficient. The reason is that we are trying to
calculate〈exp(βU−)〉whereas the Boltzmann weight factor used in the average
includes a factor exp(−βU−). The latter squeezes the high-Ucontributions
off and these contribute significantly to the average. Shing and Gubbins[37]have
formulated an efficient method combining particle insertions and removals.

10.6 Further applications and Monte Carlo methods

10.6.1 Generating ensembles of polymers

An important topic in statistical mechanics is the behaviour of polymers: long and
flexible chain-like molecules. These can be studied as a melt (a kind of liquid
consisting of polymers) or in solution. A nice review on models, theory and Monte
CarlomethodsforthisproblemcanbefoundinRef.[39]. We focus here on the
problem of a dilute polymer solution. In that case, if we can somehow model the
effect of the solvent in terms of a simple interaction, the problem reduces to studying
ensembles of individual polymers in different conformations. If the solvent is good,
then the free energy for a polymer segment which is on all sides embedded in the
solvent is lower than that of a polymer segment which is close to another polymer
segment, without solvent molecules in between them. This picture boils down to
an effective repulsion of the polymer segments.
It is useful to make a model in which the important properties of the polymer are
preserved, whereas the details of its structure on the atomic scale have disappeared
from the description. This is related to the idea of universality (seeSection 7.3.2):
only a few major features of the interactions at small length scales influence the
behaviour on longer length scales; the details do not matter. We therefore make
a ‘mesoscopic’ model of the polymer. It is a chain consisting of beads: the beads