Thermodynamics, Statistical Physics, and Quantum Mechanics

194 SOLUTIONS

The entropy Sof the gas is given by

The energyEmay now be found from

Comparing(S.4.78.7) and (S.4.78.9) gives

Note thatthis result is the same as for anultrarelativisticelectron gas

(which has the same dispersion law see Problem4.67). The total

number ofphotons isgiven by

where we let

Comparing (S.4.78.9)–(S.4.78.10) with(S.4.78.11), we canwrite

So, similarto the classicalideal gas, wehave

4.79 Dark Matter (Rutgers)

a) The virial theorem may bewrittenrelating the averagekineticenergy,
and the forces between particles(seeSect. 3.4 ofGoldstein,Classical
Mechanics):