Statistical Mechanics - Physics at Oregon State University

4.5. FERMI GAS. 83
N({n1, n2, . . .}) =
o
no
(4.62)
The energy is in general harder to find. But we now make again the assumption
that the particles are independent. This allows us to write for the energy:
E({n1, n2, . . .}) =
o
noɛo
(4.63)
where the many body energy E({n1, n2, . . .}) simply is equal to the sum of
the single particle energies ɛo corresponding to occupied states. The grand
partition function can now be written in the form
Z(T, µ, V ) =
Z(T, µ, V ) =
Z(T, µ, V ) =
1
n1=0
1
n1=0 n2=0
{n1,n2,...}
1
· · ·
e 1
k B T (µ−ɛ1)n1
e 1
k B T o (µ−ɛo)no
1
· · ·
ni=0
1
n2=0
o
e 1
k B T (µ−ɛo)no
e 1
k B T (µ−ɛ2)n2
(4.64)
(4.65)
· · · (4.66)
Since the summation variables are just dummy variables, this is equal to
Z(T, µ, V ) =
1
e 1
kB T (µ−ɛo)n
=
Zo(T, µ, V ) (4.67)
orb
n=0
where we have defined the orbital grand partition function Zo as before by:
Zo =
1
n=0
orb
µ−ɛo n k e B T (4.68)
The grand partition function for a subsystem with a single orbital is therefore
given by Zo = 1 + e µ−ɛo
kB T .
Note that there is no factor N! in front of the product in (4.67), unlike we
had before for the ideal gas partition function. The essential difference is that in
the formula above the product is over orbitals and these are distinguishable.
Previously we had a product over particles and those are identical.
Grand Energy.
The grand energy follows from
Ω(T, µ, V ) = −kBT log(Zo) = −kBT
log(Zo(T, µ, V )) (4.69)
orb