1549380323-Statistical Mechanics Theory and Molecular Simulation

416 Quantum ideal gases

the inner sum. The final sum overNcombined with the restricted sum over occupation
numbers is mathematically equivalent to an unrestricted sum over occupation num-
bers. For if we simply perform an unrestricted sum over occupationnumbers, then
all possible values ofNwill be generated automatically. Thus, we can see why the
grand canonical ensemble is preferable for fermions and bosons. The grand canonical
partition function can be written compactly as

Z(μ,V,T) =

∑

{fnm}

∏

m

∏

n

eβ(μ−εn)fnm. (11.4.6)

A second simplification results from rewriting the sum of products asa product of
sums:
∑

f 1

∑

f 2

∑

f 3

···eβ(μ−ε^1 )f^1 eβ(μ−ε^2 )f^2 eβ(μ−ε^3 )f^3 ···

=





∑

f 1

eβ(μ−ε^1 )f^1









∑

f 2

eβ(μ−ε^1 )f^2









∑

f 3

eβ(μ−ε^1 )f^3



···

=

∏

m

∏

n

∑

{fnm}

eβ(μ−εn)fnm. (11.4.7)

For fermions, each occupation-number sum contains only two terms corresponding
tofnm= 0 andfnm= 1, which yields

Z(μ,V,T) =

∏

m

∏

n

(

1 + eβ(μ−εn)

)

(Fermions). (11.4.8)

For bosons, each occupation-number sum ranges from 0 to∞and can be computed
using the sum formula for a geometric series

∑∞

n=0r

n= 1/(1−r) for 0< r <1. Thus,

eqn. (11.4.7) becomes

Z(μ,V,T) =

∏

m

∏

n

1

1 −eβ(μ−εn)

(Bosons). (11.4.9)

Note that, in each case, the summands are independent of the quantum numbermso
that we may perform the product overmvalues trivially with the result

Z(μ,V,T) =

[

∏

n

(

1 + eβ(μ−εn)

)

]g

(11.4.10)

for fermions, and

Z(μ,V,T) =

[

∏

n

1

1 −eβ(μ−εn)

]g

(11.4.11)

for bosons, whereg= (2s+ 1) is the number of eigenstates ofSˆz, which is also known
as thespin degeneracy. For spin-1/2 particles such as electrons,g= 2.