Statistical Physics, Second Revised and Enlarged Edition

Application to helium- 4 101

T T

TTTB TTTB

CV

3

2 NkkkB

3

U = 2 NkkkBT

UUU, P

UUT5/2

(a)(b)

Fig. 9. 3 The variation with temperature of the thermodynamic functions for an ideal BEgas. Compare
Fig. 8.3 for the contrastingbehaviour of an ideal FDgas. (a) Internal energyand pressure. (b) Heat capacity.

(given by (9.7)) already dumped into the ground state. And these particles do not
contribute to theinternalenergy,heat capacity, entropy, etc. The condensatefraction
isalready inits absolute-zero state.
The thermodynamic functions obtained from the occupation of the states are
illustratedinFig. 9.3(a) and(b). Thecharacteristic BE resultfor anyfunctionis:

1. 
   

   A simple power-law behaviour atT<TTTB, arising from theT^3 /^2 dependence of
   N(th)together withthe(effective) constancyofB.Thevariation ofUisT^5 /^2 ,
   and so correspondingly is the variation ofP(sinceP= 2 U/ 3 Vas shown in
   section 8.1.3). Theheat capacityCV,varies atT^3 /^2 uptoamaximum value of
   about 1. 9 NkkkBatTTTB.

   
   


2. 
   

   Aphase transition atTTTB, albeit a gentle one. In common parlance it is only
   athird-order transition, since thediscontinuityis merelyindC/dT, athird-
   orderderivative ofthefree energy;the second-orderCand the first-orderSare
   continuous.

   
   


3. 
   

   A gradualtendency to theclassicalresultwhenTTTTB,from the opposite side
   from theFDgas (compare Fig. 8.3).

   
   

Viewedfrom the perspective of hightemperatures, the onset atTTTBofthefillingof
the ground state is a sudden (and perhaps unexpected) phenomenon, like all phase
transitions. Itisreferredto as the Bose–Einstein condensation. In entropy terms,itis
an orderingink-space (i.e.in momentum space) rather than the usualtype ofphase
transition to a solid, which is an ordering in real space.

9 .2 Application to helium-4

The atom^4 He is a spin-0 boson. So just as we saw that FD statistics were relevant to

(^3) He, so we shouldexpect BE statistics to applyto (^4) He. Again, BE statistics areinthe
extreme onlya small correction for^4 Hegas, but their relevance is to the liquid.