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.