10.3 Connection to Maxwell–Boltzmann Statistics
527
Now, upon recalling [see Eq. (4.2.31)] that P V /(k B T ) = ln , we see that may
also be obtained as
(T , V , λ) = e
λz(T ,V ) .
(10.3.11a)
If we compare Eq. (4.2.4), namely
(T , V , λ) =
∞
N =0
λ
N Z N (T , V ) ,
relating the canonical partition function Z N (T , V ) to the grand partition function
, V , λ) with the expanded form of Eq. (10.3.11a), namely
(T , V , λ) =
∞
N =0
[λz(T , V )] N
N!
,
(10.3.11b)
we may conclude that Z N (T , V ) is given in terms of the single particle canonical
partition function z(T , V ) by
Z N (T , V ) =
z N (T , V )
N!
.
(10.3.12)
This procedure has provided us with a more rigorous derivation of our earlier result,
Eq. (3.2.27), for indistinguishable particles/molecules using Maxwell–Boltzmann
statistics and based upon the Gibbs rationalization for division of the power z N
by N !.
Let us now briefly reconsider the mean occupation number, n k , which is given
by
n k =
λe −ββ k
1 ± λe −ββ k
=
1
1
λ e ββ k ± 1
.
(10.3.13)
If we recall that λ = e μ/k B T , we see that λe −ββ = 1 when μ = . This result has
strong implications for each of the statistics: from Eq. (10.2.22), n k goes to infinity
for bosons, takes the value
1
2 for fermions, and takes the value 1 for the limiting
case of Maxwell–Boltzmann statistics as k approaches μ. This can also be seen
from Fig. 10.1, which compares mean occupation numbers for the three cases. We
note also that for large values of (i.e., μ) the quantum states of the system
are rarely occupied, if at all, and the probability of multiple occupancy becomes
essentially nil.