1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal boson gas 433

=−

4 V g

√

πλ^3

∫∞

0

dx x^2 ln(1−ζe−x

2

)−gln(1−ζ), (11.6.4)

where the change of variables in eqn. (11.5.2) has been made. Now, the function
ln(1−x) has the following power series expansion:

ln(1−y) =−

∑∞

l=1

yl

l

. (11.6.5)

Using eqn. (11.6.5) allows the pressure to be expressed as

Pλ^3

gkT

=

∑∞

l=1

ζl

l^5 /^2

−

λ^3

V

ln(1−ζ), (11.6.6)

and by a similar procedure, the average particle number becomes

ρλ^3

g

=

∑∞

l=1

ζl

l^3 /^2

+

λ^3

V

ζ

1 −ζ

. (11.6.7)

In eqn. (11.6.7), the term that has been split off represents the average occupation of
the ground (n= (0, 0 ,0)) state:

〈f 0 m〉=

ζ

1 −ζ

, (11.6.8)

wheref 0 m≡fn=(0, 0 ,0)m. Since〈f 0 m〉must be greater than or equal to 0, it follows that
there are restrictions on the allowed values of the fugacityζ. First, sinceζ= exp(βμ),
ζmust be positive. However, in order that the average occupation of the ground state
be positive, we must also haveζ <1. Therefore,ζ∈(0,1), so thatμ <0. The fact
thatμ <0 suggests that adding particles to the ground state is favorable,which turns
out to have fascinating consequences away from the classical limit.Before exploring
these in Section 11.6.2, however, we first treat the low-density, high-temperature limit,
where classical effects dominate.

11.6.1 Low-density, high-temperature limit

In a manner analogous to the fermion case, the low-density, high-temperature limit
can be treated using a perturbative approach. At high temperature, the fugacity is
sufficiently far from unity that the divergent terms in the pressureand density ex-
pressions can be safely neglected. Although it may not be obvious thatζ≪1 at high
temperature, recall thatζ= exp(−|μ|/kT). Moreover,μdecreases sharply in the low-
density limit, and sinceμ <0, this means|μ|is large, andζ≪1. Thus, ifζis very
different from 1, the divergent terms in eqns. (11.6.6) and (11.6.7),which have aλ^3 /V
prefactor, vanish in the thermodynamic limit.
As in the fermion case, we assume that the fugacity can be expanded as

ζ=a 1 ρ+a 2 ρ^2 +a 3 ρ^3 +···. (11.6.9)

Then, from eqn. (11.6.7), the density becomes