164 Dealingwith interactions
ofthe non-interactingidealgas. This contrasts withthe Fermiliquids,inwhichthe
simple gas theory still works accurately, so long as a few parameters are ‘adjusted’.
For the Boseliquid,thewhole temperaturedependenceismodified.The transition
to superfluidityin^4 Heisaλ-transition, rather than the muchgentler third-order
transition (compare section 11.1) at the ‘condensation temperature’of an ideal BE gas.
Itseemsthattheinteractionsenhancetheco-operativebehaviourofthesystemnearthe
transition. (Thisis reminiscent ofour remarksabout the transition toferromagnetism
in real materials which have short-range interactions (section 11.3) as opposed to
theidealmeanfield(long-range)interaction ofsection 11.2. Co-operationbetween
noninteractingbosons bystatistics onlyis an extreme example of a long-range effect.)
There is another mystery about liquid^4 He, thought to arise from the interactions.
Inthe Bose–Einsteingas andin realliquid^4 He one can explainthelow temperature
behaviour in terms of a two-fluid model(see section 9.2).AsTapproaches zero, the
whole material is seen to behave as a pure superfluid, i.e. there is no normal fluid
remaining.Asdescribedinthe previous section, the^4 Heisthermally dead.Inthe
theoretical friendly BE gas, the superfluid is pictured simply as having all particles
together in the one-particle ground state. That is the Bose–Einstein condensation.
Thusitis naturalandreasonabletovisualize thepure superfluid^4 Heinthe same way,
as pure ‘condensate’. But actually, it seems that interactions have a much more subtle
effect. Two experiments using entirelydifferent techniques (oneinvolving neutron
scatteringandtheother surface tension)have measuredthe ‘condensatefraction’inthe
pure superfluid, and agree that it is about 13–14% only of the total particle number. It
seems that theinteractions actuallydeplete the one-particle groundstate andthat most
of the^4 He atoms are scatteredinto other states,but that the occupation ofthese other
states is not observed in any normal transport or thermal experiments. A convincing
physicalexplanation ofthis strange state ofaffairs remains as yet undiscovered,
it seems.
1 4.4 Real imperfect gases
Inthelast section ofthischapter, we return to ordinaryhonestgases,liketheairwe
breathe. So far we have made two simplifications to the treatment of realgases. One
is that, for all but low-temperature helium gas, the dilute MB limit applies, i.e. the
quantum nature ofthegashas no practicaleffect. A classicaltreatment applies. This
is a verygood approximation, and one that we shall continue to use and to exploit.
The second simplification has been to ignore interactions. This is all right for dilute
enough gases, where themolecules aredistant, anditiswell known thatidealgas
theory is accurate and universal in this limit. However, it is equally well known that
gasesliquefyathighenoughdensities andlow enoughtemperatures, andthathere
interactions playthedominant role. Theidea ofstudyingthe statistics ofimperfect
gases is to see the onset of interaction effects, starting from the ideal gas side.
We shall dothisin two stages. Step oneistoformulate statisticalphysicsina
classicalmanner – thishasgreatinterest,inthat thewholesubject predatedquantum