Physical Applications
Suppose the potential energy of a gas of
n
point charges with positions
x
1
,
x
2
,
…
,
x
n
and free to move on the infinite line
-
∞
<
x
<
∞
, is given by
W
=
1
2
∑
ℓ
=
1
n
x
ℓ
2
-
∑
1
≤
ℓ
<
j
≤
n
ln
|
x
ℓ
-
x
j
|
The probability density of the positions when the gas is in thermodynamic
equilibrium is:
P
(
x
1
,
…
,
x
n
)
=
C
exp
(
-
W
(
k
T
)
)
where
k
is the Boltzmann constant,
T
the temperature and
C
a constant.
Then the partition function (with
β
=
1
(
k
T
)
) is given by
ψ
n
(
β
)
=
∫
ℝ
n
ⅇ
-
β
W
ⅆ
x
=
(
2
π
)
n
2
β
-
(
n
2
)
-
(
β
n
(
n
-
1
)
4
)
×
(
Γ
(
1
+
1
2
β
)
)
-
n
∏
j
=
1
n
Γ
(
1
+
1
2
j
β
)
For
n
charges free to move on a circular wire of radius
1
,
W
=
-
∑
1
≤
ℓ
<
j
≤
n
ln
|
ⅇ
ⅈ
θ
ℓ
-
ⅇ
ⅈ
θ
j
|
and the partition function is given by
ψ
n
(
β
)
=
1
(
2
π
)
n
∫
[
-
π
,
π
]
n
ⅇ
-
β
W
ⅆ
θ
1
⋯
ⅆ
θ
n
=
Γ
(
1
+
1
2
n
β
)
Γ
(
1
+
1
2
β
)
)
-
n
