Fluctuations in particle number for a photon gas Harvey S. Leff Citation: American Journal of Physics 83, 362 (2015); doi: 10.1119/1.4904322 View online: http://dx.doi.org/10.1119/1.4904322 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/83/4?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Onsager symmetry relations and ideal gas effusion: A detailed example Am. J. Phys. 82, 123 (2014); 10.1119/1.4827829 Comment on “The Gibbs paradox and the distinguishability of identical particles,” by M. A. M. Versteegh and D. Dieks [Am. J. Phys.79, 741–746 (2011)] Am. J. Phys. 80, 170 (2012); 10.1119/1.3657773 Dissipation, fluctuations, and conservation laws Am. J. Phys. 69, 113 (2001); 10.1119/1.1289211 Fluctuation formulas in molecular-dynamics simulations with the weak coupling heat bath J. Chem. 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However, it is shown here that the fluctuation-compressibility theorem does not hold for photons and, in fact, that fluctuations do exist. VC 2015 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4904322] I. INTRODUCTION It is therefore tempting to combine the non-existence of jT, deduced from Eq. (7), with Eq. (8) to conclude that the var- The goal of this paper is to investigate fluctuations in a iance of N does not exist.6,7 photon gas that is contained in a box of interior volume V Two specific objectives here are to show that the and wall temperature T. The emission of photons by the fluctuation-compressibility theorem does not hold for the walls generates the photon gas, and in thermodynamic equi- photon gas, and that in fact, the fluctuations in photon num- librium, fluctuations in the number of photons occur because ber are well defined. It is difficult to find a discussion of ei- of ongoing photon emission and absorption. The average ther of these points in the existing literature.8 number N of photons adjusts in accord with V and T, and h i In the subsequent sections, I first review ways to obtain the resulting pressure is solely a function of T; i.e., P P(T). average occupation numbers and corresponding variances The relevant equations of state for a photon gas of¼ volume 1 and then calculate relevant averages and show why the V and temperature T have been examined extensively. The fluctuation-compressibility theorem fails to apply to the pho- average number of photons N , pressure P(T), entropy S(T, ton gas. Following this, I discuss a Gedanken experiment V), and internal energy (averageh i total energy) U(T, V) are: that illustrates how an attempt to measure jT fails, consistent with the known non-existence of j . Brief concluding N aVT3; (1) T h i¼ remarks are in Sec. VI. 1 PT bT4; (2) ðÞ¼ 3 II. CANONICAL AND GRAND CANONICAL AVERAGES 4 ST; V bVT3; (3) ðÞ¼ 3 As preparation for the calculation of fluctuations in the number of photons in Sec. III, here I review relevant aver- 4 U T; V bVT ; (4) ages and variances and emphasize that the canonical and ð Þ¼ grand canonical ensembles give the same results. with Suppose that a photon gas has allowable single-photon 3 energies { }; i.e., is the (single-particle) energy of a pho- 16pk f 3 7 3 3 s s a ðÞ 2:03 10 mÀ KÀ (5) ton in state s. Denote the corresponding occupation numbers ¼ h3c3 ¼ Â by {ns}. Then the possible energies for the gas are and E n1; n2; … n , where the sum is over the set {s} of ð Þ¼ s s s 5 4 all single-particle states. 8p k 16 3 4 P b 7:56 10À mÀ JKÀ : (6) The canonical partition function for the photon gas is ¼ 15h3c3 ¼ Â nss=kT 1 1 n =kT In Eqs. (5) and (6), h, c, and k are Planck’s constant, the Z T; V eÀ s eÀ s s : (9) ð Þ¼ ¼ speed of light, and Boltzmann’s constant, respectively, and ns s 1 ns 0 s Xf g P Y¼ X¼ f s nnÀ is the Riemann zeta function. ð BecauseÞ¼ the pressure depends only on temperature, any The last step in Eq. (9)—replacement of a sum of products slow, isothermalP change of volume will leave the pressure by a product of sums—is understandable for a finite number unchanged, so the isothermal compressibility, M of states, i.e., when s 1,2,…M, in which case, ¼ 1 @V 1 1 1 n =kT n22=kT n11=kT jT ; (7) Z T; V eÀ M M eÀ eÀ ÀV @P T ð Þ¼ ÁÁÁ n 0 n2 0 n1 0 XM¼ X¼ X¼ 2 M does not exist. Meanwhile, the fluctuation-compressibility 1 nss=kT theorem states that3–5 eÀ : (10) ¼ s 1 n 0 Y¼ Xs¼ N2 N 2 kT h iÀ2h i jT: (8) Assuming that the interchange of sum and product in the last N ¼ V step holds in the limit M , Eq. (9) results. h i !1 362 Am. J. Phys. 83 (4), April 2015 http://aapt.org/ajp VC 2015 American Association of Physics Teachers 362 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 73.25.252.116 On: Fri, 20 Mar 2015 17:26:26 For material particles, whose number is conserved, the n2 n 2 n 1 n : (16) h s iÀh si ¼h sið þh siÞ sum over the set {ns} would carry the constraint sns N constant. Thus, the canonical partition function is Notably, the variance of n is expressible solely in terms of ¼ ¼ P s sometimes written as ZN(T,V). However, for the photon gas, the average, ns . This is curious because one expects a var- h i 2 no such constraint applies, and each occupation number ns iance to entail ns . A similar property can be corroborated can run from 0 to without constraint, so the partition func- directly for theh i variance of N using the ensemble- tion is denoted simply1 by Z(T, V). independent variance expression Notably, Z(T, V) in Eq. (9) is identical to the correspond- 2 2 ing grand canonical partition function for a photon gas 2 9,10 Z N2 N n n with zero chemical potential. To see this, group together h iÀh i ¼ s À h si all terms in Eq. (9) with n N, and then sum over all *+ s ! s ! i i ¼ X X possible N, namely from N 0 to . I insert a (cosmetic) nrns nr ns N ¼ 1 factor z in the summationP expression—with the specifica- ¼ r s ½h iÀh ih i X X tion that z 1. Here, z plays the role of fugacity in the grand n2 n 2 ¼ l=kT s s canonical ensemble, defined by z e 1; this is consist- ¼ s ½h iÀh i ent with l 0, the known chemical potential¼ for the photon X ¼ ns 1 ns : (17) gas. For each positive integer value of N with ni N, the ¼ h i½ þh i i ¼ s sum over {ni} is then formally ZN, the canonical partition X function for a fictitious system of N particles withP the photon In Eq. (17), ns is an occupation number for single-particle energy spectrum, but with N fixed. The result is that state s and should not be confused with the average occupa- tion number ns . The second line arises by writing each 1 N h i Z T; V z ZN the grand partition function: summation squared as a sum over s followed by a sum over ð Þ¼N 0 ¼Z¼ r. The third line follows because of the statistical independ- X¼ (11) ence of nr and ns, namely nrns nr ns 0 for r s, leaving only those terms withh riÀs.h Theih lasti¼ line comes6¼ Equation (11) is the standard form of the grand canonical about using Eq. (16). Evidently, it¼ is the latter statistical in- partition function. The appearance of the canonical fixed-N dependence that leads to the variance in N being dependent 2 partition function ZN arises solely from mathematical consid- only on the set nr and not on nr . erations and does not contradict the fact that actual photon In view of thefh thirdig line in Eq. (17)fh ,ig the variance in the total gases have fluctuating numbers of photons. number of photons is the sum of the variances of the occupa- Retaining the condition z 1 in the remainder of this sec- tion numbers for all the single-particle states.