We may derive the properties of a quantum gas in another way,
making use of the () ensemble in Gibbsean phase space.
Recalling the general definition of the grand partition function,
,
we now write as a sum (in place of an integral)
over states:

(5.5)

The sum
is to be taken over all permitted population numbers of all
states , again requiring that
.
The permitted values of are:
and for fermions, and
for bosons.
In this manner we arrive at

(5.6)

It is easy to show that this is equal to

(5.7)

Now we can insert the possible values of . We find that for

Fermions ():

(5.8)

and for Bosons (
):

(5.9)

Having secured the grand partition function, we can now apply the
well-known formulae for pressure, mean particle number, and internal
energy (see Section 4.3) to determine the thermodynamic
properties of the system. The mean population number of a given
state is

(5.10)

Inserting for the respective expression for Fermi or Bose particles
we once more arrive at the population densities of equ. 5.4.

In the following sections we discuss the properties of a few
particularly prominent fermion and boson gases.