Appendix F: The statistical
mechanics of
Bose–Einstein
condensation
F
F.1 The statistical mechanics
of photons 315
F.2 Bose–Einstein
condensation 316
This appendix does not attempt to reproduce the standard treatment
of a Bose–Einstein condensate found in statistical mechanics texts, but
aims to give a complementary viewpoint, that emphasises the link be-
tween photons and atoms, and also to describe BEC in a harmonic
potential.
F.1 The statistical mechanics of photons
The Planck formula for the energy density of radiation per unit band-
widthρ(ω), used in the Einstein treatment of radiation (see eqn 1.29),
can be written as a product of three factors:
ρ(ω)dω=ω×fph(ω)×Dph(ω)dω. (F.1)
Hereω is the photon energy, the functionfph(ω)=1/
(
eβω− 1
)
,
withβ=1/kBT,determines the number of photons per energy level
andDph(ω) is the density of states per unit bandwidth.^1 Although the^1 The number of states in phase-space
with wavevectors betweenkandk+dk
equals the volume of a spherical shell
of thickness dktimes the density of the
states ink-space, 4πk^2 dk×V/(2π)^3.
For photons we need an extra factor
of 2, because of the different possible
polarizations, and the substitutionk=
ω/c.
distributionfphbecomes very large asω→0 (infra-red divergence),
the integration over the frequency distribution (using the substitution
x=βω) yields a finite result for the total energy of the radiation in
the volumeV:
E=V
∫∞
0
ρ(ω)dω∝VT^4. (F.2)
This result follows from dimensional considerations, without the evalu-
ation of the definite integral.^2 This integral forEis a particular case^2 The energy density E/V has the
sameT^4 dependence as the Stefan–
Boltzmann law for the power per unit
area radiated by a black body (as
expected, sincecE/Vcorresponds to
power divided by area).
of the general expression in statistical mechanics for the energy of the
system which is obtained by summing the energy over all occupied levels:
E=
∑
i
f(εi)εi. (F.3)
Heref(εi) gives the distribution over the levels of energyεi. The integral
in eqn F.2, for the particular case of photons, gives a close approximation