Fluctuations in Particle Number for a Photon Gas Harvey S

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

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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 Fluctuations in particle number for a photon gas Harvey S. Leffa),b) Department of Physics and Astronomy, California State Polytechnic University, Pomona, California 91768 and Physics Department, Reed College, Portland, Oregon 97202 (Received 6 May 2014; accepted 3 December 2014) 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. 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 occupation 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. It is useful to ¼ tion, it is convenient to use the following notation and define the relative root-mean-square (rms) fluctuation fs for approach. As implied by Eq. (9) and used explicitly in Ref. state s, and the corresponding rms fluctuation f for the total 11, the average numbers of photons in the canonical and number of photons. These are, respectively, grand canonical ensembles, respectively, are 2 2 ns ns 1 ns @ lnZ @ ln fs h iÀh i þh i > 1; (18) ns kT and ns kT Z : (12) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin ¼ n ¼À @ h i¼À @ s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis s T;V s T;V h i h i and Because Z T; V from Eq. (11), this implies12 ð Þ¼Z 2 2 s=kT N N eÀ h iÀh i (19) ns ns : (13) f : ¼h i¼1 e s=kT qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN À À h i Given that ns ns , it follows that the variances of ns in the ¼h i Although fs > 1, no such property emerges for f, and in canonical and grand canonical ensembles are equal; i.e., fact—as we shall see—typically, f 1. 2 2 @ ns ns ns kT h i III. FLUCTUATIONS IN THE NUMBER OF h iÀh i ¼À @s T;V PHOTONS @ ns 2 2 kT ns ns : (14) To calculate the average number of photons and its var- ¼À @s ¼ À T;V iance, I first use the grand canonical ensemble with z 1 ¼ These equalities of average occupation numbers and their var- (i.e., l 0). I combine Eqs. (13), (15), and (17) and convert ¼ iances for a single-particle state in the two ensembles mean the sums over microstates to integrals following a standard 3 that I need use only one notation. In what follows I choose to technique for an assumed three-dimensional container of volume V. Note that there is no Bose condensation for a pho- retain only the grand ensemble notation for averages. 13 The average total number of photonshi can be written as ton gas, so it is not necessary to split off a term from the integral, which is necessary for a material ideal gas of bosons =kT eÀ s in a three-dimensional box. Using the abbreviations N ns : (15) s=kT h i¼ s h i¼ s 1 eÀ À 3 X X 8pV kT Meanwhile, inserting Eq. (13) into the derivative in Eq. (14) x and A ðÞ3 ; (20) gives for the variance kT hc ðÞ

363 Am. J. Phys., Vol. 83, No. 4, April 2015 Harvey S. Leff 363

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 the average number of photons is exceeds that for photons by eleven orders of magnitude, and therefore the relative fluctuation for air is much smaller. s=kT 2 x 2 2 eÀ 1 x eÀ The main point is that the variance N N exists for N A dx h iÀh i h i¼ 1 e s=kT ! 1 e x the photon gas, and for any finite temperature T the relative s À À ð0 À À X 7 3 3 3 fluctuation f vanishes in the thermodynamic limit V . A 2f 3 2:028 10 mÀ KÀ VT : (21) !1 ¼ ½ðÞ¼ ðÞÂ IV. INAPPLICABILITY OF FLUCTUATION- Similarly, converting the sum to an integral in Eq. (17) and COMPRESSIBILITY THEOREM using Eq. (21) leads to Given that the fluctuations in photon number exist, but s=kT 2s=kT 2 2 eÀ eÀ the isothermal compressibility does not, it is clear that the N N 2 s=kT =kT fluctuation-compressibility theorem, Eq. (8), fails for the h iÀh i ¼ s 1 eÀ þ 1 e s À ðÞÀ X 2 x À photon gas. To understand why, I outline a proof of the 1 x e A À dx fluctuation-compressibility theorem, modeled after Pathria’s 2 3 ! 0 1 e x proof for material particles (not photons). ð ðÞÀ À 1 2 7 3 3 3 Consider a macroscopic, open sub-volume of material gas Ap 2:776 10 mÀ KÀ VT ¼ 3 ¼ ðÞÂ particles embedded within a larger gas. Particles can freely flow into and out of this volume; i.e., is variable. Because 1:369 N : (22) N ¼ h i z el=kT is a variable, kT @=@l z @=@z , and thus ¼ ð ÞT;V ¼ ð ÞT;V An alternative procedure is to write the variance expression in Eq. (24) can be written as N2 N 2 kT @ N ln @ h iÀ2h i 2 h i : (26) N z Z (23) N ¼ N @l T;V h i¼ @z T;V "#z 1 h i h i ¼ The remainder of the proof proceeds by assuming that V is and fixed, but the volume per particle v V= N is variable. The right side of Eq. (26) can be written as h i 2 2 @ N N N z h i : (24) 2 h iÀh i ¼ @z T;V kT @ N v @ V=v "#z 1 h i kT ðÞ ¼ 2 N @l T;V ¼ V @l T;V h i With this procedure, I first assume general z 1, take the v 6¼ =kT kT @ needed z derivatives of ln ln 1 zeÀ s and Z¼À s ð À Þ ¼ÀV @l N , then set z 1, and finally, convert the sums to integrals. T;V hThisi leads, once¼ again, to Eqs. (21) andP (22). kT @v @P Clearly, the average and variance given by Eqs. (21) and À : (27) ¼ V @P ; @l (22) both exist for finite T and V and are of the same order of T VT;V magnitude. Using Eqs. (21) and (22) in Eq. (19), the relative Finally, a more useful expression for @P=@l T;V can be rms fluctuation in N is obtained using the Gibbs-Duhem equation,ð Þ 4 3=2 3=2 2:597 10À m K 1:170 dl v dP s dT; (28) f Â : (25) ¼ À ¼ pVT3 ¼ N h i where v and s are the volume and entropy per particle. It fol- 2 2 1 Equations (22) andffiffiffiffiffiffiffiffi(25) show thatpffiffiffiffiffiffiffiffiN N is propor- lows from Eq. (28) that @P=@l vÀ , and therefore ð ÞT;V ¼ tional to N , and the relative fluctuationh if Àish proportionali to Eqs. (26) and (27) lead to the fluctuation-compressibility 1= N .h Thesei same properties hold for material gases that theorem: satisfyh thei fluctuation-compressibility theorem, Eq. (8). For example,pffiffiffiffiffiffiffiffi applying Eq. (8) to air, treated as a classical ideal N2 N 2 kT 1 @v kT : gas with 1 1 , and one finds h iÀ2h i jT jT =P; kT=V jT = N air N ¼ V À v @P T;V ¼ V 2 2¼ ð Þ ¼ h i N N N , or fair 1= N . More gener- h i h iair Àh iair ¼h iair ¼ h iair ally, Eq. (8) can be written as 2 2 This completes the proof, which I emphasize holds for a ma- N pNffiffiffiffiffiffiffiffiffiffiffi N intensive thermodynamic variable. h iÀh i ¼h iÂ terial gas. Returning to the ideal gas, to gain a sense of what this However, for a photon gas, the proof above fails. Equation means numerically, consider a room with dimensions 3 m (26) must be evaluated at l 0 and thus has no remaining l ¼ 4m 2:5 m and thus V 60 m3. At a typical room tempera-Â dependence. Thus, the steps in Eq. (27) that were used for tureÂ of 300 K, the number¼ density of photons is N =V material particles cannot be executed. Further, in the Gibbs- 5:5 1014 m–3 and the total photon numberh i ¼ is Duhem equation (28), v and P are independent of l, so the Â 16 1 N 3:3 10 . In contrast, the number density of air mol- expression, @P=@l T;V vÀ that was useful for the mate- heculesi¼ at theÂ same temperature and atmospheric pressure is rial gas doesð not hold.Þ In¼ fact, for the photon gas P is not a 25 –3 N air=V 2:5 10 m and the total number of molecules function of l and Eq. (28) reduces to dP/dT s/v S/V. The h i ¼ Â 27 ¼ ¼ is N air 1:5 10 . The relative rms fluctuation for pho- latter equation is consistent with the result obtained by dif- h i ¼ Â 9 tons and air, respectively, are f 5:5 10À and ferentiating Eq. (2) and comparing the result with Eq. (3) but 14 ¼ Â fair 2:6 10À . The average number of air molecules is of no help with the proof being attempted. ¼ Â

364 Am. J. Phys., Vol. 83, No. 4, April 2015 Harvey S. Leff 364

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 Other proofs4,5 of Eq. (8) fail similarly for the photon gas, a)Electronic mail: [email protected] b)Present address: 12705 SE River Road, Portland, Oregon 97222. and the conclusion is that the standard proofs cannot be used 1 for the photon gas. Moreover, there cannot exist any other 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 intro- proof because, as shown explicitly in Sec. III, for the photon ductory physics,” Am. J. Phys. 70, 792–797 (2002) and references therein. gas the variance of N deﬁnitely does exist, and as shown in 2Because P is solely a function of T for the photon gas, @P=@V 0. ð ÞT ¼ Sec. I, the isothermal compressibility jT does not exist. Given this, it is tempting to conclude that @V=@P is inﬁnitely large, but ð ÞT Clearly, Eq. (8) does not hold for the photon gas. 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,ð it is correctÞ ¼ toð say thatÞ

@V=@P T (and thus jT) does not exist. 3ð Þ V. A GEDANKEN EXPERIMENT R. K. Pathria, Statistical Mechanics, 2nd ed. (Butterworth-Heinemann, Oxford, 1996), pp. 100–101. The non-existence of the isothermal compressibility can 4R. Kubo, Statistical Mechanics (North Holland-Interscience-Wiley, New York, 1965), pp. 398–399. be understood at least in part by envisaging a Gedanken 5 experiment where the photon gas is contained within a verti- K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1986), pp. 152–153. cal cylinder in a gravitational field. A floating (frictionless) 6For example, see Ref. 3, p. 172, where after demonstrating that the average piston is the container’s ceiling, and the walls are maintained number of photons is proportional to VT3, it is written that the latter result at temperature T. Begin with the piston fixed such that the “cannot be taken at its face value because in the present problem the mag- container volume is V. The number of photons N adjusts, nitude of the fluctuations in the variable N, which is determined by the 1 and the equilibrium pressure P(T) is establishedh ini accord- quantity @P=@V À , is infinitely large.” 7It is worthð pointingÞ out that fluctuations in energy are well defined for a ance with Eq. (2). If the piston is released so that it can float, 2 3 photon gas: the variance in energy is kT Cv, and Cv VT (see, e.g., Ref. 3, thermal equilibrium exists only if the piston weight provides p. 101). This suggests that fluctuations in photon number/ also exist, contrary an external pressure equal to P(T). to what one might expect from Eqs. (7) and (8). This dichotomy provides To measure the compressibility, add an arbitrarily light further incentive to clarify that the fluctuations in N do indeed exist. grain of sand to the piston. One might hope to calculate an 8An anonymous reviewer of this manuscript has kindly informed me that a approximate value of the compressibility using calculation of the fluctuations in N for the photon gas is presented in C. 1 Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: jT VÀ DV=DP. However, with the walls at fixed temperature,À the equilibrium pressure of the photon gas does not Introduction to Quantum Electrodynamics (Wiley, New York, 1989), pp. 235–236. change and the extra sand grain causes the piston to drop 9Thermal photons are known to have zero chemical potential. See, for precipitously to the container floor; i.e., the unstable photon example, R. Baierlein, “The elusive chemical potential,” Am. J. Phys. 69, gas collapses to zero volume. During the collapse, the photon 423–434 (2001). Two compelling ways to argue that l 0 for thermal gas follows an irreversible path through non-equilibrium photons are: (i) the empirically observed distribution of photon¼ frequencies states. Once equilibrium is re-established, there are zero pho- for blackbody radiation agrees with the prediction for a quantum ideal tons in a zero-volume container. bose gas of photons only if the chemical potential l is set equal to zero; and (ii) using Eqs. (3) and (4) to obtain S(U,V,N) (4=3)b1=4V 1=4U3=4, Thus, an arbitrarily small change in does lead to a ¼ P not application of the thermodynamic identity l T @S=@N then gives ¼ ð ÞU;V correspondingly small DV and there is no way to approxi- l 0. 10 ¼ mate the isothermal compressibility. The fact that a measure- A justification sometimes given for l 0 is that the photon number is not ¼ ment of the isothermal compressibility jT is not possible is fixed, but, rather, is indefinite. See, for example, F. Herrmann and P. W€urfel, “Light with nonzero chemical potential,” Am. J. Phys. 73, consistent with the non-existence of jT established on theo- retical grounds in Sec. I. 717–721 (2005). These authors observe that the indefinite number of photons alone is not sufficient to conclude that l 0 because particle numbers are not conserved in chemical reactions, where¼ the material constituents have nonzero chemical potential. Indeed, the zero chemical potential result VI. CONCLUDING REMARKS holds only for thermal photons. For example, if light is in “chemical” equi- Because all matter radiates, photons are ubiquitous in the librium with the excitations of matter whose chemical potential is nonzero—e.g., the electron-hole pairs in a light emitting diode—then the universe. The oldest photons, those in the cosmic microwave chemical potential of the light must be nonzero too. The full argument, background radiation, go back to the big bang. In this which entails recognition that lelectron lhole l, is given by Herrmann respect, photons are indeed special. The photon gas is special and W€urfel. See also P. W€urfel, “Theþ chemical¼ potential of radiation,” too in that it is a relatively simple quantum mechanical, rela- J. Phys. C Solid State Phys. 15, 3967–3985 (1982). tivistic, and thermal model, as evidenced by the occurrence 11F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Secs. 9.3–9.7. of the fundamental constants h, c, and k in Eqs. (1)–(6). 12 Because photons can be, and are, continually created and This quantum statistics formula for Bose-Einstein single-particle state occupation numbers {ns} can also be found using a counting argument. annihilated by matter, their total number in a closed box fluc- See, for example, D. ter Haar, Elements of Statistical Mechanics (Holt, tuates continually. Those fluctuations are finite, and the sug- Rinehart and Winston, New York, 1964), Secs. 4.1–4.3. gestion that the variance of N does not exist because the 13See, e.g., J. Honerkamp, Statistical Physics: An Advanced Approach with isothermal compressibility doesh i not exist is incorrect. I have Applications (Springer, Berlin, 1998), pp. 228–233. To see why, note that shown here that specific assumptions used for a material gas the single-photon energy is shc=V1=3, where V L3 for an assumed s ¼ ¼ to prove the fluctuation-compressibility theorem, namely the cubical volume, with s s2 s2 s2, and where s , s , and s run over x þ y þ z x y z proportionality of the variance of N with the isothermal the positive integers. The ground state g has sx sy sz 1 and sg p3. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ ¼ ¼ compressibility, do not hold for photons.h i Using the result that the average total number of photons N V, it fol- 2=3 h i/ ffiffiffi lows that the ratio ng = N VÀ 0 in the thermodynamic limit V ; i.e., the fractionh i ofh photonsi/ in the! ground (or any other single) state !1 ACKNOWLEDGMENTS is zero. Note: This argument requires that we set l 0 before taking the thermodynamic limit, which is the correct order. If we¼ were to take (incor- The author grateful to Raj Pathria and Paul Beale for their rectly) the thermodynamic limit first, we would mistakenly “discover” an helpful comments on a first draft of this article. actually nonexistent singularity for the ground (or any other) state.

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