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,
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:
This can also be expressed using the fugacity by noting that
Thus,