1549380323-Statistical Mechanics Theory and Molecular Simulation

434 Quantum ideal gases

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 +···. (11.6.10)

By equating like powers ofρon both sides, the coefficientsa 1 ,a 2 ,a 3 ,...can be deter-
mined as for the fermion gas. Working to first order inρgives

a 1 =

λ^3

g

, ζ≈

λ^3 ρ

g

, (11.6.11)

and the equation of state is the expected classical result

P

kT

=ρ. (11.6.12)

Working to second order, we find

a 2 =−

λ^6

23 /^2 g^2

ζ=

λ^3 ρ

g

−

λ^6

23 /^2 g^2

ρ^2 , (11.6.13)

and the second-order equation of state becomes

P

kT

=ρ−

λ^3

25 /^2 g

ρ^2. (11.6.14)

The second virial coefficient can be read off and is given by

B 2 (T) =−

1

25 /^2 g

λ^3 =−

0. 1768

g

λ^3 < 0. (11.6.15)

In contrast to the fermion case, the bosonic pressure decreases from the classical value
as a result of spin statistics. Thus, there appears to be an “effective attraction” between
the particles. Unlike the fermion gas, where the occupation numbers of the available
energy levels are restricted by the Pauli exclusion principle, any number of bosons
can occupy a given energy state. Thus, at temperatures slightly lower than those at
which a classical description is valid, particles can “condense” into lower energy states
and cause small deviations from a strict Maxwell-Boltzmann distribution of kinetic
energies.

11.6.2 The high-density, low-temperature limit

At high density, the work needed to insert an additional particle intothe system be-
comes large. Sinceμmeasures this work andμ <0, the high-density limit is equivalent
to theμ→0 or theζ→1 limit. In this limit, the full problem, including the divergent
terms, must be solved: