Statistical Mechanics - Physics at Oregon State University

100 CHAPTER 5. FERMIONS AND BOSONS
4 1
f 1 (λ) = √
2 π λ
∞
x 2 e
dx
−x2
λ−1 + e−x22 0
(5.53)
which is clearly always positive.
If we look at equation(5.48) we see that, since the right hand side is always
increasing, there are some simple relations between λ and the temperature T .
For T = 0 the quantum density goes to zero, and hence the left hand side of
equation(5.48) goes to ∞. The right hand side has to go to infinity, too, and
this means that λ goes to infinity. If the temperature goes to infinity, the left
hand side of the equation goes to zero, which means that λ goes to zero. For
values in between, λ decreases when the temperature increases! Hence, for a
fixed value of N, we have:
Low temperature expansions.
lim λ = ∞ (5.54)
T →0
lim λ = 0 (5.55)
T →∞
∂λ
∂T
V,N
< 0 (5.56)
The goal is to find the chemical potential as a function of the number of
particles, at a given temperature and volume. We need to use equation (5.48)
to do that. We have seen that for a given density low temperatures correspond
to values of λ going to infinity. Therefore, we need to investigate the behavior
of the function f 3 for large arguments. We start with the integral form, since
2
we are outside the range of the series expansion:
which is equal to
4
f 3 (λ) = √
2 π
4
f 3 (λ) = √
2 π
∞
x 2 dx
Using y = x 2 and defining λ = e ν we get
2
f 3 (λ) = √
2 π
Integration by parts gives via:
0
4
f 3 (λ) = √
2 3π
e −x2
λ −1 + e −x2
∞
x 2 1
dx
λ−1ex2 + 1
0
∞
0
∞
0
(5.57)
(5.58)
1
ey−ν √
ydy (5.59)
+ 1
1
ey−ν 3
dy 2 (5.60)
+ 1