1549380323-Statistical Mechanics Theory and Molecular Simulation

van der Waals equation 173

Suppose next thatU 0 andU 1 are both pair-wise additive potentials of the form

U 0 (r 1 ,...,rN) =

∑N

i=1

∑N

j>i

u 0 (|ri−rj|)

U 1 (r 1 ,...,rN) =

∑N

i=1

∑N

j>i

u 1 (|ri−rj|). (4.7.25)

By the same analysis that led to eqn. (4.6.45), the unperturbed average ofU 1 is

〈U 1 〉 0 = 2πNρ

∫∞

0

dr r^2 u 1 (r)g 0 (r), (4.7.26)

whereg 0 (r) is the radial distribution function of the unperturbed system at agiven
density and temperature. In this case, the Helmholtz free energy, to first order inU 1 ,
is

A(N,V,T)≈−

1

β

ln

(

Z(0)(N,V,T)

N!λ^3 N

)

+ 2πρN

∫∞

0

dr r^2 u 1 (r)g 0 (r). (4.7.27)

We now wish to use the framework of perturbation theory to formulate a statistical
mechanical model capable of describing real gases and a gas–liquid phase transition. In
Fig. 4.2(a), we depicted a pair-wise potential energy capable of describing both gas and
liquid phases. However, the form of this potential, eqn. (3.14.3), is too complicated for
an analytical treatment. Thus, we seek a crude representation of such a potential that
can be treated within perturbation theory. Consider replacing the 4 ǫ(σ/r)^12 repulsive
wall by a simplerhard spherepotential,

u 0 (r) =

{

0 r > σ

∞ r≤σ

, (4.7.28)

which we will use to define the unperturbed ensemble. Since we are interested in the
gas–liquid phase transition, we will work in the low density limit appropriate for the
gas phase. In this limit, we can apply eqn. (4.6.75) and write the unperturbed radial
distribution function as

g 0 (r)≈e−βu^0 (r)=

{

1 r > σ

0 r≤σ

=θ(r−σ), (4.7.29)

whereθ(x) is the Heaviside step function. For the perturbationu 1 (r), we need to
mimic the attractive part of Fig. 4.2(a), which is determined by the− 4 ǫ(σ/r)^6 term.
In fact, the particular form ofu 1 (r) is not particularly important as long asu 1 (r)< 0
for allrandu 1 (r) is short-ranged. Thus, our crude representation of Fig. 4.2(a)is
shown in Fig. 4.6. Despite the simplicity of this model, some very interesting physics
can be extracted.
Consider the perturbative correctionA(1), which is given to first order inU 1 by

A(1)≈ 2 πNρ

∫∞

0

r^2 u 1 (r)g 0 (r) dr