1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal fermion gas 421

11.5.2 The high-density, low-temperature limit

The high-density, low-temperature limit exhibits the largest departure from classi-
cal behavior. Using eqn. (11.5.3), we obtain the following integral expression for the
density:

ρλ^3 =

4 g

√

π

∫∞

0

x^2 dx

ζ−^1 e−x^2 + 1

. (11.5.19)

Starting with this expression, we can derive an expansion in the inverse powers of
lnζ≡μ/kT, as these inverse powers will become decreasingly small asT→0, allowing
the leading order behavior to be deduced. We begin by introducing the variable

ν= lnζ=

μ

kT

(11.5.20)

and developing an expansion in its inverse powers. We will sketch out briefly how this
is accomplished. We first introduce a change of variabley=x^2 , from whichx=

√

y
anddx=dy/(2

√

y). When this change is made in eqn. (11.5.19), we obtain

ρλ^3 =

2 g

√

π

∫∞

0

√

ydy

ey−ν+ 1

. (11.5.21)

The integral can be carried out by parts using

u=

1

ey−ν+ 1

, du=−

1

(ey−ν+ 1)^2

ey−νdy

dv=y^1 /^2 dy, v=

2

3

y^3 /^2 , (11.5.22)

which gives

ρλ^3 =

4 g

3

√

π

∫∞

0

y^3 /^2 ey−νdy

(ey−ν+ 1)^2

. (11.5.23)

Next, we expandy^3 /^2 abouty=ν:

y^3 /^2 =ν^3 /^2 +

3

2

ν^1 /^2 (y−ν) +

3

8

ν−^1 /^2 (y−ν)^2 +···. (11.5.24)

This expansion is now substituted into eqn. (11.5.23) and the resulting integrals over
yare performed, which yields

ρλ^3 =

4 g

3

√

π

[

(lnζ)^3 /^2 +

π^2

8

(lnζ)−^1 /^2 +···

]

+O(1/ζ), (11.5.25)

where the fact thatμ/kT≫1 has been used for the low temperature limit. The high
density limit implies a high chemical potential, which makesζ(ρ) = eβμ(ρ)large as
well. A largeζalso helps ensure the convergence of the series in eqn. (11.5.25), since
the error falls off with powers of 1/ζ.