1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal boson gas 435

Pλ^3

gkT

=

∑∞

l=1

ζl

l^5 /^2

−

λ^3

V

ln(1−ζ)

ρλ^3

g

=

∑∞

l=1

ζl

l^3 /^2

+

λ^3

V

ζ

1 −ζ

. (11.6.16)

We will need to refer to the two sums in eqns. (11.6.16) often in this section, so let us
define them as follows:

g 3 / 2 (ζ) =

∑∞

l=1

ζl

l^3 /^2

g 5 / 2 (ζ) =

∑∞

l=1

ζl

l^5 /^2

. (11.6.17)

Thus, eqns. (11.6.16) can be expressed as

Pλ^3

gkT

=g 5 / 2 (ζ)−

λ^3

V

ln(1−ζ) (11.6.18)

ρλ^3

g

=g 3 / 2 (ζ) +

λ^3

V

ζ

1 −ζ

. (11.6.19)

First, consider eqn. (11.6.19) for the density. The termζ/(1−ζ) diverges atζ= 1. It
is instructive to ask about the behavior ofg 3 / 2 (ζ) atζ= 1. In fact,g 3 / 2 (1), given by

g 3 / 2 (1) =

∑∞

l=1

1

l^3 /^2

, (11.6.20)

is a special type of a mathematical function known as aRiemann zeta-function. In
general, the Riemann zeta-functionR(n) is defined to be

R(n) =

∑∞

l=1

1

ln

(11.6.21)

(values ofR(n) are provided in many standard math tables). The quantityg 3 / 2 (1) =
R(3/2) is a pure number whose approximate value is 2.612. Moreover, from the form
ofg 3 / 2 (ζ), it is clear that, sinceζ <1,g 3 / 2 (1) is the maximum value ofg 3 / 2 (ζ). A plot
ofg 3 / 2 (ζ) is given in Fig. 11.2.
The figure also indicates that the derivativeg 3 ′/ 2 (ζ) diverges atζ= 1, despite the
value of the function being finite. Sinceζ <1, it follows that

g 3 / 2 (ζ)< g 5 / 2 (ζ). (11.6.22)

It is possible to solve eqn. (11.6.19) forζby noting that unlessζis very close to 1,
the divergent term must vanish in the thermodynamic limit as a result of theλ^3 /V