Statistical Mechanics - Physics at Oregon State University

84 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
and the product has become a sum as expected. The energy should be the
sum of the energies of the subsystems, and this implies a product in the grand
partition function since the energy always needs the logarithm of this grand
partition function.
Once the grand energy is available, all thermodynamic variables can be obtained.
For example, the number of particles follows from
or
N = −
as expected.
∂Ω
∂µ T,V
N =
= kBT
orb
orb
e µ−ɛo
k B T
1 + e µ−ɛo
k B T
Entropy of a system of Fermions.
1
Zo(T, µ, V )
∂Zo(T, µ, V )
∂µ
T,V
(4.70)
=
fF D(ɛo; T, µ) (4.71)
A useful formula for the entropy expresses the entropy in terms of the distribution
functions.
S = −
∂Ω
∂T µ,V
With the help of:
we get
orb
= kB log(Zo(T, µ, V )) + kBT
orb
∂ log(Zo(T, µ, V ))
∂T
orb
µ,V
(4.72)
S = kB
orb
log(1 + e µ−ɛo
k B T ) + kBT
e µ−ɛ
k B T = 1
e ɛ−µ
k B T
=
orb
1
e ɛ−µ
k B T + 1 − 1
1
1
fF D − 1 = fF D
1 − fF D
S = kB log(1 +
orb
fF
D
) − kB
1 − fF D
orb
S = −kB log(1 − fF D) − kB
orb
fF D
1−fF D
e µ−ɛo
k B T
1 + e µ−ɛo
k B T
1 + fF D
1−fF D
orb
=
fF D log(
µ − ɛo
−kBT 2
fF D
(4.73)
(4.74)
log( ) (4.75)
1 − fF D
fF D
) (4.76)
1 − fF D