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.

## References

Because *P* is solely a function of *T* for the photon gas, $(\u2202P/\u2202V)T=0$. Given this, it is tempting to conclude that $(\u2202V/\u2202P)T$ is infinitely large, but this is incorrect. The reason is that the identity $(\u2202V/\u2202P)T=1/(\u2202P/\u2202V)T$ does not hold when the left side is zero. Rather, it is correct to say that $(\u2202V/\u2202P)T$ (and thus *κ _{T}*) does not exist.

For example, see Ref. 3, p. 172, where after demonstrating that the average number of photons is proportional to *VT*^{3}, 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 $(\u2202P/\u2202V)\u22121$, is infinitely large.”

It is worth pointing out that fluctuations in energy *are* well defined for a photon gas: the variance in energy is *kT*^{2}*C _{v}*, and

*C*∝

_{v}*V T*

^{3}(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.

*N*for the photon gas is presented in

*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)

*b*

^{1∕4}

*V*

^{1∕4}

*U*

^{3∕4}, application of the thermodynamic identity $\mu =T(\u2202S/\u2202N)U,V$ then gives

*μ*= 0.

*μ*= 0 is that the photon number is not fixed, but, rather, is indefinite. See, for example,

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 $\mu electron+\mu hole=\mu $, is given by Herrmann and Würfel.

*n*} can also be found using a counting argument. See, for example,

_{s}*V*=

*L*

^{3}for an assumed cubical volume, with $s\u2261sx2+sy2+sz2$, and where

*s*,

_{x}*s*, and

_{y}*s*run over the positive integers. The ground state

_{z}*g*has

*s*= 1 and $sg=3$. Using the result that the average total number of photons $\u3008N\u3009\u221dV$, it follows that the ratio $\u3008ng\u3009/\u3008N\u3009\u221dV\u22122/3\u21920$ 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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