432 Quantum ideal gases
CV
〈N〉k=
π^2 kT
2 εF. (11.5.78)
From eqn. (11.5.77), the pressure can be obtained immediately as
P=
2
5
ρεF[
1 +
5
12
π^2(
kT
εF) 2
+···
]
, (11.5.79)
which constitutes a low-temperature equation of state.
11.6 The ideal boson gas
The behavior of the ideal boson gas is dramatically different from that of the ideal
fermion gas. Indeed, bosonic systems have received considerableattention in the liter-
ature because of a phenomenon known asBose–Einstein condensation, which we will
derive in this section.
As with the fermion case, the treatment of the ideal boson gas begins with the
equations for the pressure and average particle number in terms of the fugacity:
PV
kT=−g∑
nln(
1 −ζe−βεn)
(11.6.1)
〈N〉=g∑
nζe−βεn
1 −ζe−βεn. (11.6.2)
Careful examination of eqns. (11.6.1) and (11.6.2) reveals an immediate problem: The
termn= (0, 0 ,0) diverges for both the pressure and the average particle number as
ζ→1. These terms need to be treated carefully, hence we split them offfrom the rest
of the sums in eqns. (11.6.1) and (11.6.2), which gives
PV
kT=−g∑
n′ln( 1 −ζe−βεn)−gln(1−ζ)〈N〉=g∑
n′ ζe
−βεn
1 −ζe−βεn
+gζ
1 −ζ. (11.6.3)
Here,
∑′
means that then= (0, 0 ,0) term is excluded. With these divergent terms
written separately, we can take the thermodynamic limit straightforwardly and convert
the remaining sums to integrals as was done in the fermion case. For the pressure, we
obtain
PV
kT=−g∫
dnln(
1 −ζe−βεn)
−gln(1−ζ)=− 4 πg∫∞
0dn n^2 ln(
1 −ζe−^2 π(^2) β ̄h (^2) n (^2) /mL 2 )
−gln(1−ζ)