The fluctuation-compressibility theorem of statistical mechanics states that fluctuations in particle number are proportional to the isothermal compressibility. Given that the compressibility of a photon gas does not exist, this seems to suggest that fluctuations in photon number similarly do not exist. However, it is shown here that the fluctuation-compressibility theorem does not hold for photons and, in fact, that fluctuations do exist.

1.
A detailed discussion and summary of the thermodynamic properties of a photon gas can be found in
H. S.
Leff
, “
Teaching the photon gas in introductory physics
,”
Am. J. Phys.
70
,
792
797
(
2002
) and references therein.
2.

Because P is solely a function of T for the photon gas, (P/V)T=0. Given this, it is tempting to conclude that (V/P)T is infinitely large, but this is incorrect. The reason is that the identity (V/P)T=1/(P/V)T does not hold when the left side is zero. Rather, it is correct to say that (V/P)T (and thus κT) does not exist.

3.
R. K.
Pathria
,
Statistical Mechanics
, 2nd ed. (
Butterworth-Heinemann
,
Oxford
,
1996
), pp.
100
101
.
4.
R.
Kubo
,
Statistical Mechanics
(
North Holland-Interscience-Wiley
,
New York
,
1965
), pp.
398
399
.
5.
K.
Huang
,
Statistical Mechanics
, 2nd ed. (
Wiley
,
New York
,
1986
), pp.
152
153
.
6.

For example, see Ref. 3, p. 172, where after demonstrating that the average number of photons is proportional to VT3, it is written that the latter result “cannot be taken at its face value because in the present problem the magnitude of the fluctuations in the variable N, which is determined by the quantity (P/V)1, is infinitely large.”

7.

It is worth pointing out that fluctuations in energy are well defined for a photon gas: the variance in energy is kT2Cv, and CvV T3 (see, e.g., Ref. 3, p. 101). This suggests that fluctuations in photon number also exist, contrary to what one might expect from Eqs. (7) and (8). This dichotomy provides further incentive to clarify that the fluctuations in N do indeed exist.

8.
An anonymous reviewer of this manuscript has kindly informed me that a calculation of the fluctuations in N for the photon gas is presented in
C.
Cohen-Tannoudji
,
J.
Dupont-Roc
, and
G.
Grynberg
,
Photons and Atoms: Introduction to Quantum Electrodynamics
(
Wiley
,
New York
,
1989
), pp.
235
236
.
9.
Thermal photons are known to have zero chemical potential. See, for example,
R.
Baierlein
, “
The elusive chemical potential
,”
Am. J. Phys.
69
,
423
434
(
2001
). Two compelling ways to argue that μ = 0 for thermal photons are: (i) the empirically observed distribution of photon frequencies for blackbody radiation agrees with the prediction for a quantum ideal bose gas of photons only if the chemical potential μ is set equal to zero; and (ii) using Eqs. (3) and (4) to obtain S(U,V,N) = (4∕3)b1∕4V1∕4U3∕4, application of the thermodynamic identity μ=T(S/N)U,V then gives μ = 0.
10.
A justification sometimes given for μ = 0 is that the photon number is not fixed, but, rather, is indefinite. See, for example,
F.
Herrmann
and
P.
Würfel
, “
Light with nonzero chemical potential
,”
Am. J. Phys.
73
,
717
721
(
2005
).

These authors observe that the indefinite number of photons alone is not sufficient to conclude that μ = 0 because particle numbers are not conserved in chemical reactions, where the material constituents have nonzero chemical potential. Indeed, the zero chemical potential result holds only for thermal photons. For example, if light is in “chemical” equilibrium with the excitations of matter whose chemical potential is nonzero—e.g., the electron-hole pairs in a light emitting diode—then the chemical potential of the light must be nonzero too. The full argument, which entails recognition that μelectron+μhole=μ, is given by Herrmann and Würfel.

See also
P.
Würfel
, “
The chemical potential of radiation
,”
J. Phys. C Solid State Phys.
15
,
3967
3985
(
1982
).
11.
F.
Reif
,
Fundamentals of Statistical and Thermal Physics
(
McGraw-Hill
,
New York
,
1965
), Secs. 9.3–9.7.
12.
This quantum statistics formula for Bose-Einstein single-particle state occupation numbers {ns} can also be found using a counting argument. See, for example,
D.
ter Haar
,
Elements of Statistical Mechanics
(
Holt, Rinehart and Winston
,
New York
,
1964
), Secs. 4.1–4.3.
13.
See, e.g.,
J.
Honerkamp
,
Statistical Physics: An Advanced Approach with Applications
(
Springer
,
Berlin
,
1998
), pp.
228
233
. To see why, note that the single-photon energy is ϵs=shc/V1/3, where V = L3 for an assumed cubical volume, with ssx2+sy2+sz2, and where sx, sy, and sz run over the positive integers. The ground state g has sx = sy = sz = 1 and sg=3. Using the result that the average total number of photons NV, it follows that the ratio ng/NV2/30 in the thermodynamic limit V →∞; i.e., the fraction of photons in the ground (or any other single) state is zero. Note: This argument requires that we set μ = 0 before taking the thermodynamic limit, which is the correct order. If we were to take (incorrectly) the thermodynamic limit first, we would mistakenly “discover” an actually nonexistent singularity for the ground (or any other) state.
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