Statistical Mechanics - Physics at Oregon State University

4.1. INTRODUCTION. 71
1
Fermi
Dirac
0.5
0
mu
2 k T
energy
Figure 4.1: Fermi Dirac distribution function.
lim
ɛ→∞ fF D(ɛ; T, µ) = 0 (4.10)
lim
ɛ→−∞ fF D(ɛ; T, µ) = 1 (4.11)
fF D(ɛ = µ; T, µ) = 1
2
(4.12)
and the horizontal scale for this function is set by the product kBT . In the limit
T → 0 the Fermi-Dirac function becomes a simple step function, with value 1
for ɛ < µ and value 0 for ɛ > µ. Note that at ɛ = µ the value remains 1
2 !
Bosons.
In the case of Bosons the summation in the grand partition function goes
from zero to infinity and we have
Zo(T, µ, V ) =
∞
n=0
e n(µ−ɛo)
k B T =
1
e µ−ɛo
k B T − 1
(4.13)
which is only valid when µ < ɛo, or else the series diverges. This is an important
difference with the previous case, where µ could take all values. Here the possible
values of the chemical potential are limited! The average number of particles
follows from
which gives
∂ log(Zo)
< no >= kBT
∂µ
T,V
(4.14)