4.1 Canonical Ensemble: Closed Systems
177
We may solve this equation for (N − n)/n as
N − n
n
= e
1
2 βE f .
Upon neglecting n relative to N, we arrive at the approximation
n = Ne
−
1
2 βE f ,
for n.
If we now return to expression (4.1.16), we can rewrite it in the form
U − T S = −k B T ln Z = A .
(4.1.17)
This relates the Helmholtz energy A to the partition function and the temperature
T in a very direct fashion. Equivalently, we may rewrite Eq. (4.1.17) to obtain the
partition function Z in terms of the Helmholtz energy A as
Z = e
−βA .
(4.1.18)
This expression directly relates the canonical partition function to a single thermodynamic state function (in this case, the Helmholtz energy), and possibly the
temperature. When we are able to find a thermodynamic state function that is related
to the partition function for a particular ensemble in such a simple manner, we say
that that state function is the ‘characteristic function’ for the particular ensemble.
The Helmholtz energy function is thus known as the characteristic state function for
the canonical ensemble.
Let us now reconsider the entropy and its statistical mechanical interpretation.
We might well ask, for example, ‘What happens to S when T → 0?’. To see
what does happen, we note that T → 0 is the same thing as β → ∞, which means
that summations of the type
r e −βE r become simply one term, namely e −βE 0 ,
so that the partition function reduces also to a single term, the internal energy U
becomes E 0 , and the entropy becomes k B ln 0 , or
Z
→
T →0
0 e
−βE 0 ,
(4.1.19)
U
→
T →0
U 0 = E 0 ,
(4.1.20)
S
→
T →0
k B ln 0 .
(4.1.21)
Note that the final of these three expressions tells us that at a temperature of absolute
zero the entropy is a constant, independent of all parameters characterizing the
system. This is nothing other than a version of the Third Law of Thermodynamics.