Chapter 4
Nonideal gases
In this chapter we consider gases of shortrange interaction. The “shortrange” is defined by
that the integration of the potential between two particles over the space converges, i.e., the
potential decays faster than
r
−
3
for
r
being the distance between the two particles. With the
gravitational force neglected, the interaction between neutral particles decays at least as fast
as
r
−
6
. So the shortrange interaction condition is satisfied. The case of electrons which carry
charges is rather complicated.
If the Coulomb potential is screened, the interaction is still
shortranged. But there are cases where the longrange interaction is important, especially for
electrons in lowdimensional space and at low density, such as in the case of Wigner lattices
formed by 2dimensional electron gases at low density.
4.1
Cluster expansion for nonideal classical gases
We first consider a classical gas in grand canonical ensemble. For simplicity, we assume that
there is only twobody interaction and the interaction has translational symmetry. The potential
between two particles has the form
u
i
,
j
=
u
(
r
i
−
r
j
)
,
(4.1)
which is shortranged, i.e.,
lim

r
i
−
r
j
→∞

r
i
−
r
j

3
u
i
,
j
=
0
.
(4.2)
The partition function is
Ξ =
∞
∑
N
=
0
1
N
!
ζ
N
1
h
3
N
∫
d
p
1
d
p
2
· · ·
d
p
N
∫
d
r
1
d
r
2
· · ·
d
r
N
e
−
β
(
p
2
1
+
p
2
2
+
···
+
p
2
N
)
/
(2
m
)
−
β
∑
i
<
j
≤
N
u
i
,
j
,
(4.3)
where
ζ
≡
e
βµ
is the fugacity. Following the calculation for ideal gases, the integration over
momentum can be done to obtain
Ξ =
∞
∑
N
=
0
z
N
N
!
∫
d
r
1
d
r
2
· · ·
d
r
N
∏
i
<
j
≤
N
e
−
β
u
i
,
j
,
(4.4)
where
z
≡
λ
−
3
T
exp(
βµ
) with
λ
T
≡
~
√
2
πβ/
m
being the thermal de Brogile wavelength.
47
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
48
CHAPTER 4. NONIDEAL GASES
The integration over the real space coordinates is nontrivial as the interaction potential
couples all of them. To proceed, we have to expand the integrations in order of some small
quantity. A standard way is the cluster expansion which is done with respect to the density of
the gas.
Noticing that
e
−
β
u
i
,
j
→
1 quickly as the distance between the two particles increases, we
single out the contribution due to the interaction by defining
e
−
β
u
i
,
j
≡
1
+
f
i
,
j
.
(4.5)
Thus we can expand
∏
i
<
j
≤
N
e
−
β
u
i
,
j
=
1
+
∑
f
i
,
j
+
∑
(
i
,
j
)
,
(
k
,
l
)
f
i
,
j
f
k
,
l
+
· · ·
.
(4.6)
For example,
e
−
β
(
u
1
,
2
+
u
2
,
3
+
u
3
,
1
)
=
(
1
+
f
1
,
2
) (
1
+
f
2
,
3
) (
1
+
f
3
,
1
)
=
1
+
f
1
,
2
+
f
2
,
3
+
f
3
,
1
+
f
1
,
2
f
2
,
3
+
f
2
,
3
f
3
,
1
+
f
3
,
1
f
1
,
2
+
f
1
,
2
f
2
,
3
f
3
,
1
.
The point of such expansion is that we can integrate over the variables separately if they are not
“connected”. For example, in the integration of
f
1
,
2
, the variables
r
1
and
r
2
can be separated
from the other ones. The “connection” is defined as:
1. Two coordinates
i
and
j
are connected if the term contains
f
i
,
j
;
2. Two coordinates
i
and
j
are connected if both of them are connected to another coordinate
k
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 DrKwong
 Statistical Mechanics, partition function, CLASSICAL GASES

Click to edit the document details