section 4.4
The starting point we use for the Gibbs-Duhem integration is the well
known freezing point of monodisperse hard spheres [56,39]. However, the
initial slope at 
=0 given by eqn.
4.10 is not known
here. Moreover, it cannot be calculated directly in a simulation
because both 
and 
are equal to zero in the monodisperse
limit, although the ratio 
is expected to be finite.
The second moment 
can be calculated if the composition
is known. The composition in turn is related to the
chemical potential by
where 
is a collection of terms taken as independent of
, and 
is the residual chemical
potential. With eqn.
4.3, the chemical potential
difference function can now be written as
or,
Here, 
is the
difference in residual chemical potential between a particle with
diameter 
and a particle of the reference component. This
difference can be measured in a simulation of a pure 
substance by performing `test enlargements', in which a randomly
chosen particle is enlarged from diameter 
to a random
diameter 
. One tabulates the frequency with which such moves
result in no overlap, although the moves themselves are never
accepted. This overlap probability yields the residual chemical
potential according to
where the brackets indicate the ensemble average; 
is zero
or unity, respectively, corresponding to the absence or presence of
overlap. This `ghost-growing' procedure is very similar to the Widom 'ghost' particle
insertion technique [36]. In practice, we tabulate the distance
from a randomly selected particle to its nearest neighbor; this
histogram of nearest distances is then easily converted into the
overlap histogram just described.
For small values of 
the measured residual chemical
potential might be approximated by a quadratic function
Substituting this in eqn. 4.13 results for the composition
where C is independent of 
. Because 
is a Gaussian
function, the desired second moment of the composition 
can be
analytically obtained
The latter approximation is correct to second order in 
. The
initial slope can now be written as
where 
is the difference between
the coefficient expressions evaluated in both phases. In sum, we
measure the residual chemical potential in a simulation of a pure hard
sphere system using eqn.
4.14, we fit it to a quadratic
function for small 
and use the coefficients in eqn.
4.18 to obtain the initial slope for the Gibbs-Duhem
integration.
section 4.4
