Partition function

Consider two canonical systems, 1 and 2, with particle numbers and , volumes and and at temperature T. The systems are in chemical contact, meaning that they can exchange particles. Furthermore, we assume that and so that system 2 is a particle reservoir. The total particle number and volume are

The total Hamiltonian is

If the systems could not exchange particles, then the canonical partition function for the whole system would be

where

However, and are not fixed, therefore, in order to sum over all microstates, we need to sum over all values that can take on subject to the constraint . Thus, we can write the canonical partition function for the whole system as

where is a function that weights each value of for a given N. Thus,

f(0,N) is the number of configurations with 0 particles in and N particles in .

f(1,N) is the number of configurations with 1 particles in and N-1 particles in .

etc.

Determining the values of amounts to a problem of counting the number of ways we can put N identical objects into 2 baskets. Thus,

f(0,N)=1

f(1,N)=N=N!/1!(N-1)!

f(2,N)=N(N-1)/2=N!/2!(N-2)!

etc.

or generally,

which is clearly a classical degeneracy factor. If we were doing a purely classical treatment of the grand canonical ensemble, then this factor would appear in the sum for Q(N,V,T), however, we always include the ad hoc quantum correction 1/N! in the expression for the canonical partition function, and we see that these quantum factors will exactly cancel the classical degeneracy factor, leading to the following expression:

which expresses the fact that, in reality, the various configurations are not distinguishable from each other, and so each one should count with equal weighting. Now, the distribution function is given by

which is chosen so that

However, recognizing that , we can obtain the distribution for immediately, by integrating over the phase space of system 2:

where the prefactor has been introduced so that

and amounts to the usual ad hoc quantum correction factor that must be multiplied by the distribution function for each ensemble to account for the identical nature of the particles. Thus, we see that the distribution function becomes

Recall that the Hemlholtz free energy is given by

Thus,

or

But since and , we may expand:

Therefore the distribution function becomes

Dropping the ``1'' subscript, we have

We require that be normalized:

Now, we define the grand canonical partition function

Then, the normalization condition clearly requires that

Therefore PV is the free energy of the grand canonical ensemble, and the entropy is given by

We now introduce the fugacity defined to be

Then, the grand canonical partition function can be written as

which allows us to view the grand canonical partition function as a function of the thermodynamic variables , V, and T.

Other thermodynamic quantities follow straightforwardly:

Energy:

Average particle number:

This can also be expressed using the fugacity by noting that

Thus,