The ideal fermion gas 419Pλ^3
gkT=
∑∞
l=1(−1)l+1ζl
l^5 /^2ρλ^3
g=
∑∞
l=1(−1)l+1ζl
l^3 /^2, (11.5.7)
whereρ=〈N〉/V is the number density. Although we cannot solve these equations
to obtain a closed form for the equation of state, two interesting limits can be worked
out to a very good approximation, which we examine next.
11.5.1 The high-temperature, low-density limit
Solving forζas a function of〈N〉is equivalent to solving forζas a function ofρ.
Hence, in the low-density limit, we can take an ansatz forζ=ζ(ρ) in the form of a
power series:
ζ(ρ) =a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···. (11.5.8)
How rapidly this series converges depends on how low the density actually is. Writing
out the first few terms in the pressure and density equations, we have
Pλ^3
gkT=ζ−ζ^2
25 /^2+
ζ^3
35 /^2−
ζ^4
45 /^2+···
ρλ^3
g=ζ−ζ^2
23 /^2+
ζ^3
33 /^2−
ζ^4
43 /^2+···. (11.5.9)
Substituting eqn. (11.5.8) into eqns. (11.5.9) gives
ρλ^3
g= (a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···)−1
23 /^2
(a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···)^2+
1
33 /^2
(a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···)^3 +···. (11.5.10)Eqn. (11.5.10) can be solved perturbatively, equating like powers ofρon both sides.
For example, if we work only to first order inρ, then we have
ρλ^3
g=a 1 ρ ⇒ a 1 =λ^3
g⇒ζ≈λ^3 ρ
g. (11.5.11)
When eqn. (11.5.11) is substituted into eqn. (11.5.9) for the pressure and only terms
first order in the density are kept, we obtain
Pλ^3
gkT=
ρλ^3
g⇒
P
kT=ρ=〈N〉
V
, (11.5.12)
which is just the classical ideal gas equation. If we now go out to second order inρ,
eqn. (11.5.9) gives