Thermodynamics, Statistical Physics, and Quantum Mechanics

182 SOLUTIONS

The condition for Bose condensation tooccur isthat, at someparticular

temperature, thechemicalpotentialgoes tozero. Then the number of

particles outside the Bose condensatewill bedetermined by the integral

Thisintegral should convergesinceNis a givennumber.Expanding around

in order todetermineconditions forconvergence of the integralyields

So, this integraldiverges at andthere is noBosecondensation
for thisregion. (Forinstance, in two dimensions, particleswithordinary
dispersion law wouldnot Bose-condense.) Inthreedimensions,
so thatBosecondensationdoesoccur.

4.72 How Hot the Sun? (Stony Brook)

(See Problem 2 of Chapter 4 in Kittel and Kroemer, Thermal Physics.) The
distribution ofphotonsover the quantum states with energy
is given by Planck’s distribution (the Bose–Einstein distributionwith
chemicalpotential

where is the temperature of the radiation which we consider equal to the
temperature of the surface of the Sun. To find the total energy, we can
replace the sum overmodes by an integral over frequencies:

where the factor 2accounts for the two transversephotonpolarizations.
The energy of radiation in an interval andunitvolume istherefore