172
4 Mean Values and Thermodynamics
We have seen how to evaluate E from z. We shall now see that we may also
evaluate E 2 in much the same way. Let us start with the definition of E 2 , viz.,
E
2
≡
r
p r E
2
r ,
which becomes
E
2
=
1
z
∂ 2 z
∂β 2
V
upon substitution of p r from Eq. (3.2.12a) into the defining relation, followed by
simplification of the resultant expression. If we now introduce this result as well as
that for E into the defining relation for the variance, we find that
((E)
2
=
1
z
∂ 2 z
∂β 2
V
−
1
z 2
∂z
∂β
2
V
(4.1.4a)
=
∂
∂β
1
z
∂z
∂β
V
=
∂
∂β
∂ ln z
∂β
V
,
which can be further simplified to
((E)
2
=
∂ 2 ln z
∂β 2
V
.
(4.1.4b)
As for the transition from canonical ensemble expression (4.1.2b) for the singleparticle internal energy u to the N -particle expression (4.1.2c) for U , we see that
the single-particle canonical ensemble expression (4.1.4a) passes directly over into
the N -particle canonical ensemble expression
((E)
2
N =
∂ 2 ln Z N
∂β 2
V
.
(4.1.4c)
Let us now examine the statistical mechanical expression for the heat capacity at
constant volume, C V , starting from its definition in terms of the internal energy U .
We obtain
C V ≡
∂U
∂T
V
=
∂β
∂T
∂U
∂β
V
= −
1
k B T 2
∂
∂β
−
∂ ln Z N
∂β
V
,