Statistical Mechanics - Physics at Oregon State University

5.1. FERMIONS IN A BOX. 95
non-analytic behavior. The physics of such behavior is a phase transition! Hence
if we do not discuss convergence issues properly, we will miss many important
phase transitions! For example, if we replace sums by integrals for bosons, we
see no Bose-Einstein condensation.
Evaluating the integral.
Our task is to evaluate:
1
V Ω(T, µ, V ) = −(2S + 1)(2π)−3
kBT
d 3 k log(1 + λe − 2 k 2
2Mk B T ) (5.23)
The integral is changed into a simple form by a coordinate transformation.
Define
and we obtain
2
x =
2MkBT
1
V Ω(T, µ, V ) = −(2S + 1)(2π)−3 − 2
kBT
2MkBT
3
2
After integrating over the angular variables we find
1
V Ω(T, µ, V ) = −(2S+1)(2π)−3 − 2
kBT
2MkBT
3
2
4π
Next we introduce the thermal wavelength
2 2π
λT =
MkBT
1
2
k (5.24)
1
2
∞
0
d 3 x log(1 + λe −x2
)
(5.25)
x 2 dx log(1+λe −x2
)
(5.26)
(5.27)
which is approximately the de-Broglie wavelength of a particle with energy
kBT . Again, this combination is introduced for historical reasons to simplify
the formulas. The result is a small difference from the purist definition of a
de-Broglie wavelength. Note that nQ(T ) = λ −3
T . At the quantum density a
cube of dimension de-Broglie wavelength contains one particle. In other words,
the average distance between particles is the de-Broglie wavelength, and it is no
surprise that quantum effects will be important!
We also introduce a function f 5 (λ) by
2
∞
4
f 5 (λ) = √
2 π
0
x 2 dx log(1 + λe −x2
) (5.28)