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(Chris Devlin) #1

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

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