Statistical Physics, Second Revised and Enlarged Edition

104 Bose–Einsteingases

oftheflow tube. The source reservoir warms upasitisdepletedofsuperfluidwhereas

the receiving reservoir cools down as the superfluid enters.

9 .3 Phoneybosons

Abosonis a particleofspin 0,1,2,...by definition. Ourdiscussion sofarinthe
chapter has been based on an idealgas of ‘real’ particles of massM.However,there
is another case, which is that the bosons could be particles with no rest mass. The
obvious exampleisaphotongas. An evacuatedboxis never ‘empty’,inthatitwill
contain electromagnetic radiation in thermal equilibrium with the walls of the box.
Such thermal equilibrium radiation is called black-body radiation, and it may best
be consideredasaBEgas, anidealgas ofphotons. The secondanalogous example
is that the lattice vibrations of a solid can be modelled similarly as a gas of sound
waves, or ‘phonons’,inabox whichistheboundary ofthesolid.
The newfeature ofbothofthesegasesisthat the particles ofthegas are massless.
This has the importance that thenumberof the gas particles is not fixed. The particles
maybe, andare,destroyedandcreatedcontinually. Thereis no conservation ofNin
the macrostate andthesystemisdefinednotby(N,V,T)but simply by(V,T).The
empty box will come to equilibrium with a certain average number and energy of
photons (or phonons) whicharedependent onlyonthe temperature andthevolume
ofthebox. So when wederive thethermalequilibriumdistributionfor thegas, there
is a change. The usual number restriction

∑

ni=N,(5.7), does not enter, and the

derivation of the distribution (section 5.4.2) is to be followed through without it. The

answer is almost self-evident.

Since there is noNrestriction,there is noα(so also noB,and no chemical

potentialμ). But the rest ofthederivationisasbefore, andwe obtainthe ‘modified

BE’distribution

f(ε)= 1 /[exp(−βε)− 1 ] (9.9)

withas usualβ=− 1 /kkkBT.We shall now applythis result to photons and phonons.

9 .3.1 Photons and black-body radiation

This is an important topic. The radiation may be characterized by its ‘spectral density’,

u(ν),definedsuchthatu(ν)δνisthe energy oftheradiation withfrequenciesbetween

νandν+δν.This spectraldensitywas studiedexperimentallyandtheoretically

in the last century and was the spur for Planck in the early 1900s to first postulate

quantization, anidea then taken upbyEinstein. ThePlanckradiationlaw, ofgreat

historicalandpracticalimportance, canbe readily derivedusingtheideas ofthis

chapter, as we shall now see.

ThemodifiedBEdistribution (9.9) tellsusthe number ofphotons per state of

energyεinthermalequilibrium at temperatureT.In order to calculate thespectral