Statistical Mechanics - Physics at Oregon State University

96 CHAPTER 5. FERMIONS AND BOSONS
The notation 5
2 will become clear in a moment. This function was carefully
analyzed, an important occupation before the days of computers when series
expansions were badly needed to get numbers. Even now they are important
since we can often use simple limit forms of these analytical functions in model
calculations. The grand energy is
Ω(T, µ, V ) = − (2S + 1)
spin
degeneracy
V
simple
volume
dependence
kBT
energy
scale
λ −3
T
volume
scale
f 5
2 (λ)
density
effects
(5.29)
The right hand side is independent of volume, depends on µ only via f 5 (λ) and
2
on T in three places.
Density dependence.
The function f 5 (λ) has some simple properties. For large values of x one
2
can expand the logarithm and the integrant is approximately equal to x 2 λe −x2
Therefore, the integral is well behaved at the upper limit of x → ∞ and we are
allowed to play all kinds of games with the integrant. If λ < 1 (remember that
λ > 0) the logarithm can be expanded in a Taylor series for all values of x and
since the integral is well behaved it is allowed to interchange summation and
integration. Using
|z| < 1 ⇒ log(1 + z) =
∞
n=1
1
n zn (−1) n
and generalizing λ to be able to take on complex values, we have
|λ| < 1 ⇒ log(1 + λe −x2
) =
∞
n=1
1
n (−1)n λ n e −nx2
.
(5.30)
(5.31)
and since the convergence of the series is uniform we can interchange summation
and integration. This leads to
4
|λ| < 1 ⇒ f 5 (λ) = √
2 π
where we have used
∞
n=1
∞
0
1
n (−1)n λ n
y 2 dye −ny2
∞
0
x 2 dxe −nx2
= 1√
1
π
4 n √ n
=
∞ λn (−1) n+1
n=1
n 5
2
(5.32)
(5.33)
This defines why we used the notation 5
2 . In general, one defines a family of
functions fα(z) by