Statistical Mechanics - Physics at Oregon State University

3.4. GRAND PARTITION FUNCTION. 53
∂Ω
Next we use the following thermodynamical relations, ∂µ = −N and
V,T
∂Ω
∂β = −T β
V,µ
−1
∂Ω
∂T V,µ = T Sβ−1 to get
∂f
U = Ω − f + µN − β + T S (3.48)
∂β
and with U = Ω + T S + µN this gives
∂f
0 = f + β =
∂β µ,V
or
µ,V
∂βf
∂β µ,V
(3.49)
f(T, V ) = kBT g(V ) (3.50)
The partial derivative of the grand potential with respect to T yields the entropy:
∂Ω
S = −
∂T µ,V
∂kBT log(Z)
=
+ kBg(V )
∂T V,µ
(3.51)
In the limit T → 0 the grand partition function approaches log(g0) where g0 is
the degeneracy of the ground state. Therefore, the first term on the right hand
side approaches kB log(g0), which is the entropy at zero temperature. Therefore,
lim g(V ) = 0 (3.52)
T →0
but, since g(V ) does not depend on T, this shows that g(V ) = 0 and hence
or
Ω = −kBT log(Z) (3.53)
Z = e −βΩ
(3.54)
This form is very similar to the one we found for the relation between the
canonical sum and the Helmholtz free energy.
Working with the grand canonical ensemble is necessary in every situation
where the number of particles is allowed to vary. Besides for systems in diffusive
contact, it is also the ensemble of choice in quantum field theory, where particles
are destroyed and created via operators a and a † and the quantum states are
elements of a Fock space with variable numbers of particles.
Once we have found the grand potential, we are also able to evaluate the
entropy. A simple formula for the entropy is again obtained in terms of the
probabilities of the quantum states. We have
S =
U − Ω − µN
T
= 1
T
s
ɛsP rob(s) + kB log(Z) − µ
T
nsP rob(s) (3.55)
s