2Classical gases
II Statistical Physics
2.1 The classical partition function
In the canonical ensemble, the quantum partition function is
Z =
X
n
e
−βE
n
.
What is the classical analogue? Classically, we can specify the state of a system
by a point in phase space, which is the space of all positions and momentum.
For example, if we have a simple particle, then a point in phase space is just
(
q
(
t
)
, p
(
t
)), the position and momentum of the particle. It is conventional to use
q
instead of
x
when talking about a point in the phase space. So in this case,
the phase space is a 6-dimensional space.
The equation of motion determines the trajectory of the particle through
phase space, given the initial position in phase space. This suggests that we
should replace the sum over states by an integral over phase space. Classically,
we also know what the energy is. For a single particle, the energy is given by
the Hamiltonian
H =
p
2
2m
+ V (q).
So it seems that we know what we need to know to make sense of the partition
function classically. We might want to define the partition function as
Z
1
=
Z
d
3
q d
3
p e
−βH(p,q)
.
This seems to make sense, except that we expect the partition function to be
dimensionless. The solution is to introduce a quantity
h
, which has dimensions
of length times momentum. Then we have
Definition
(Partition function (single particle))
.
We define the single particle
partition function as
Z
1
=
1
h
3
Z
d
3
q d
3
p e
−βH(p,q)
.
We notice that whenever we use the partition function, we usually differentiate
the log of
Z
. So the factor of
h
3
doesn’t really matter for observable quantities.
However, recall that entropy is just given by
log Z
, and we might worry that
the entropy depends on
Z
. But it doesn’t matter, because entropy is not actually
observable. Only entropy differences are. So we are fine.
The more careful reader might worry that our choice to integrate e
−βH(p,q)
against d
3
q
d
3
p
is rather arbitrary, and there is no good a priori reason why we
shouldn’t integrate it against, say, d
3
q d
3
p
5
instead.
However, it is possible to show that this is indeed the “correct” partition
function to use, by taking the quantum partition function, and then taking the
limit ~ → 0. Moreover, we find that the correct value of h should be
h = 2π~.
We will from now on use this value of
h
. In the remainder of the chapter, we
will mostly spend our time working out the partition function of several different
systems as laid out in the beginning of the chapter.