10.8 The Partition Function for Ensembles
465
P k = exp (−1 + α) exp (−βE k + γ N k ) .
(10.109)
As before for the canonical ensemble, the constraint
k P k = 1 determines α, and
we write
P k =
1
exp (−βE k + γ N k ) ,
(10.110)
where the ‘grand canonical partition function’ is defined as
V , γ ) ≡
k
exp (γ N k − βE k )
(10.111)
=
k
λ
N k exp (−βE k ) ,
(10.112)
with λ ≡ exp (γ ) and the sum is taken over all the ensembles, recognizing that the
ensemble number k has a distinct value for each of the possible quantum states of
the system being considered.
If we group all the ensemble members that have the same number of particles,
and label these groups by the index l, then
V , γ ) =
∞
N =0
λ
N
l
exp (−βE N,l ) =
N
λ
N Z(β, V , N)
(10.113)
with Z(β, V , N) being the canonical partition function for a system with a fixed
number of particles N . Using an ideal gas as a heat bath, as in Sect. 10.5.3, we will
have again that β = 1/(k B T ).
A note of caution: The expression for the grand canonical partition function in
Eq. (10.113) starts with N = 0. The canonical partition function Z(β, V , 0) need
not be unity. As the canonical partition function contains a factor g 0 exp (−βE 0 )
from the selected choice for the zero of energy for all systems,
Z(β, V , 0) = g 0 exp (−βE 0 ) .
Not taking this zero-point energy into account will make the sizes of the terms in
Eq. (10.113) for N > 0 incorrect relative to the N = 0 term.
With our new grand canonical partition function , the entropy of the system
becomes
S = −k B
k
P k ln P k = −k B
k
P k ((−βE k + γ N k ) − ln )
= k B βE − k B γ N + k B ln ,
(10.114)