1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal boson gas 437

0 1 2 3

vg/l

3

z

1

1/R(3/2)

Fig. 11.3Plot of eqn. (11.6.28).

Forρλ^3 /g < R(3/2),ζwill be different from 1 by more than 1/V, and the diver-
gent term proportional toζ/(1−ζ) can, therefore, be safely neglected. Thus, for
ρλ^3 /g < R(3/2), we only need to solveρλ^3 /g=g 3 / 2 (ζ) forζ. Combining these re-
sults, the general solution forζvalid at high density and low temperature can be
expressed as

ζ=







1 − λ

(^3) /V
(ρλ^3 /g)−R(3/2)
ρλ^3
g > R(3/2)
root ofg 3 / 2 (ζ) =ρλ
3
g
ρλ^3
g < R(3/2)

, (11.6.27)

which in the thermodynamic limit becomes

ζ=







1 ρλ

3

g > R(3/2)

root ofg 3 / 2 (ζ) =ρλ

3

g

ρλ^3

g < R(3/2)

. (11.6.28)

A plot ofζvs.vg/λ^3 =V g/〈N〉λ^3 is shown in Fig. 11.3. According to the figure,
the pointR(3/2) is special, asζundergoes a transition there to the (approximately)
constant value of 1.
In order to see what the effect of this transition has on the average occupation
numbers, recall that the latter can be determined using

〈N〉=

∑

n,m

ζe−βεn

1 −ζe−βεn

=

∑

n,m

〈fnm〉, (11.6.29)