1549380323-Statistical Mechanics Theory and Molecular Simulation

The ideal boson gas 441

V

P

T 1

T 2

T 3

Fig. 11.5Plot of the isotherms of the equation of state in eqn. (11.6.43). HereT 1 > T 2 > T 3.
The dotted line connects the transition points from constant to decreasing pressure and is of
the formP∼V−^5 /^3.

the classical ideal gas. This is likewise in contrast to the fermion idealgas, where
asT →0, the pressure remains finite. For the boson gas, asT →0, the pressure
vanishes, in keeping with the notion of an effective “attraction” between the particles
that causes them to condense into the ground state, which is a state of zero energy.
Other thermodynamic quantities follow from the equation of state.The energy can
be obtained fromE= 3PV/2, yielding

E=







3

2

kTV

λ^3 g^5 /^2 (1) ρ > ρ^0 ,T < T^0

3

2

kTV

λ^3 g^5 /^2 (ζ) ρ < ρ^0 ,T > T^0

, (11.6.44)

and the heat capacity at constant volume is obtained fromCV = (∂E/∂T)v, which
gives

CV

〈N〉k

=







15

4

g 5 / 2 (1)

ρλ^3 T < T^0

15

4

g 5 / 2 (ζ)

ρλ^3 −

9

4

g 3 / 2 (ζ)

g 1 / 2 (ζ) T > T^0

. (11.6.45)

The plot of the heat capacity in Fig. 11.6 exhibits a cusp atT=T 0. Experiments
carried out on liquid^4 He, which has been observed to undergo Bose–Einstein con-
densation at aroundT=2.18 K, have measured an actual discontinuity in the heat
capacity at the transition temperature, suggesting that Bose–Einstein condensation is
a phase transition known as aλtransition. By contrast, the heat capacity of the ideal