Fluctuations in particle number for a photon gas

Transcription

1 Fluctuation in particle number for a photon ga Harvey S. Leff Citation: American Journal of Phyic 83, 362 (2015); doi: / View online: View Table of Content: Publihed by the American Aociation of Phyic Teacher Article you may be intereted in Onager ymmetry relation ideal ga effuion: A detailed example Am. J. Phy. 82, 123 (2014); / Comment on The Gibb paradox the ditinguihability of identical particle, by M. A. M. Verteegh D. Diek [Am. J. Phy.79, (2011)] Am. J. Phy. 80, 170 (2012); / Diipation, fluctuation, conervation law Am. J. Phy. 69, 113 (2001); / Fluctuation formula in molecular-dynamic imulation with the weak coupling heat bath J. Chem. Phy. 113, 2976 (2000); / Fluctuation in the number of particle of the ideal ga: A imple example of explicit finite-ize effect Am. J. Phy. 67, 1149 (1999); / Thi article i copyrighted a indicated in the article. Reue of AAPT content i ubject to the term at: Downloaded to IP:

2 Fluctuation in particle number for a photon ga Harvey S. Leff a),b) Department of Phyic Atronomy, California State Polytechnic Univerity, Pomona, California Phyic Department, Reed College, Portl, Oregon (Received 6 May 2014; accepted 3 December 2014) The fluctuation-compreibility theorem of tatitical mechanic tate that fluctuation in particle number are proportional to the iothermal compreibility. Given that the compreibility of a photon ga doe not exit, thi eem to ugget that fluctuation in photon number imilarly do not exit. However, it i hown here that the fluctuation-compreibility theorem doe not hold for photon, in fact, that fluctuation do exit. VC 2015 American Aociation of Phyic Teacher. [ I. INTRODUCTION The goal of thi paper i to invetigate fluctuation in a photon ga that i contained in a box of interior volume V wall temperature T. The emiion of photon by the wall generate the photon ga, in thermodynamic equilibrium, fluctuation in the number of photon occur becaue of ongoing photon emiion aborption. The average number hni of photon adjut in accord with V T, the reulting preure i olely a function of T; i.e., P ¼ P(T). The relevant equation of tate for a photon ga of volume V temperature T have been examined extenively. 1 The average number of photon hni, preure P(T), entropy S(T, V), internal energy (average total energy) U(T, V) are: with hni ¼aVT 3 ; (1) PT ð Þ ¼ 1 3 bt4 ; (2) ST; ð VÞ ¼ 4 3 bvt3 ; (3) UðT; VÞ ¼bVT 4 ; (4) a ¼ 16pk3 fðþ 3 h 3 c 3 ¼ 2: m 3 K 3 (5) b ¼ 8p5 k 4 15h 3 c 3 ¼ 7: m 3 JK 4 : (6) In Eq. (5) (6), h, c, k are Planck contant, the peed of light, Boltzmann contant, repectively, fðþ ¼ P n n i the Riemann zeta function. Becaue the preure depend only on temperature, any low, iothermal change of volume will leave the preure unchanged, o the iothermal compreibility, j T 1 ; (7) T doe not exit. 2 Meanwhile, the fluctuation-compreibility theorem tate that 3 5 ¼ kt V j T: (8) It i therefore tempting to combine the non-exitence of j T, deduced from Eq. (7), with Eq. (8) to conclude that the variance of N doe not exit. 6,7 Two pecific objective here are to how that the fluctuation-compreibility theorem doe not hold for the photon ga, that in fact, the fluctuation in photon number are well defined. It i difficult to find a dicuion of either of thee point in the exiting literature. 8 In the ubequent ection, I firt review way to obtain average occupation number correponding variance then calculate relevant average how why the fluctuation-compreibility theorem fail to apply to the photon ga. Following thi, I dicu a Gedanken experiment that illutrate how an attempt to meaure j T fail, conitent with the known non-exitence of j T. Brief concluding remark are in Sec. VI. II. CANONICAL AND GRAND CANONICAL AVERAGES A preparation for the calculation of fluctuation in the number of photon in Sec. III, here I review relevant average variance emphaize that the canonical gr canonical enemble give the ame reult. Suppoe that a photon ga ha allowable ingle-photon energie { }; i.e., i the (ingle-particle) energy of a photon in tate. Denote the correponding occupation number by {n }. Then the poible energie for the ga are Eðn 1 ; n 2 ; Þ ¼ P n, where the um i over the et {} of all ingle-particle tate. The canonical partition function for the photon ga i ZðT; VÞ ¼ X fn g e P n =kt ¼ Y1 X 1 ¼1 n ¼0 e n =kt : (9) The lat tep in Eq. (9) replacement of a um of product by a product of um i undertable for a finite number M of tate, i.e., when ¼ 1,2, M, in which cae, ZðT; VÞ ¼ X1 n M ¼0 ¼ YM X 1 ¼1 n ¼0 e n M M =kt X1 e n 2 2 =kt X1 e n 1 1 =kt n 2 ¼0 n 1 ¼0 e n =kt : (10) Auming that the interchange of um product in the lat tep hold in the limit M!1, Eq. (9) reult. 362 Am. J. Phy. 83 (4), April VC 2015 American Aociation of Phyic Teacher 362 Thi article i copyrighted a indicated in the article. Reue of AAPT content i ubject to the term at: Downloaded to IP:

3 For material particle, whoe number i conerved, Pthe um over the et {n } would carry the contraint n ¼ N ¼ contant. Thu, the canonical partition function i ometime written a Z N (T,V). However, for the photon ga, no uch contraint applie, each occupation number n can run from 0 to 1 without contraint, o the partition function i denoted imply by Z(T, V). Notably, Z(T, V) in Eq. (9) i identical to the correponding gr canonical partition function Z for a photon ga with zero chemical potential. 9,10 To ee thi, group together all term in Eq. (9) with P i n i ¼ N, then um over all poible N, namely from N ¼ 0 to 1. I inert a (cometic) factor z N in the ummation expreion with the pecification that z ¼ 1. Here, z play the role of fugacity in the gr canonical enemble, defined by z e l=kt ¼ 1; thi i conitent with l ¼ 0, the known chemical potential for the photon ga. For each poitive integer value of N with P i n i ¼ N, the um over {n i } i then formally Z N, the canonical partition function for a fictitiou ytem of N particle with the photon energy pectrum, but with N fixed. The reult i that ZðT; VÞ¼ X1 N¼0 z N Z N ¼Z¼the gr partition function: (11) Equation (11) i the tard form of the gr canonical partition function. The appearance of the canonical fixed-n partition function Z N arie olely from mathematical conideration doe not contradict the fact that actual photon gae have fluctuating number of photon. Retaining the condition z ¼ 1 in the remainder of thi ection, it i convenient to ue the following notation approach. A implied by Eq. (9) ued explicitly in Ref. 11, the average number of photon in the canonical gr canonical enemble, repectively, are n ¼ hn i¼ Becaue ZðT; VÞ ¼Zfrom Eq. (11), thi implie 12 n ¼hn i¼ : (12) e =kt 1 e =kt : (13) Given that n ¼hn i, it follow that the variance of n in the canonical gr canonical enemble are equal; i.e., hn 2 i hn i 2 ¼ ¼ n ¼ 2 n2 : (14) Thee equalitie of average occupation number their variance for a ingle-particle tate in the two enemble mean that I need ue only one notation. In what follow I chooe to retain only the gr enemble notation hifor average. The average total number of photon can be written a hni ¼ X hn i¼ X e =kt 1 e =kt : (15) Meanwhile, inerting Eq. (13) into the derivative in Eq. (14) give for the variance hn 2 i hn i 2 ¼hn ið1 þhn iþ : (16) Notably, the variance of n i expreible olely in term of the average, hn i. Thi i curiou becaue one expect a variance to entail hn 2 i. A imilar property can be corroborated directly for the variance of N uing the enembleindependent variance expreion *! X + 2! 2 ¼ n hn i ¼ X r ¼ X ¼ X X X ½hn r n i hn r ihn iš ½hn 2 i hn i 2 Š hn i½1 þhn iš: (17) In Eq. (17), n i an occupation number for ingle-particle tate hould not be confued with the average occupation number hn i. The econd line arie by writing each ummation quared a a um over followed by a um over r. The third line follow becaue of the tatitical independence of n r n, namely hn r n i hn r ihn i¼0 for r 6¼, leaving only thoe term with r ¼. The lat line come about uing Eq. (16). Evidently, it i the latter tatitical independence that lead to the variance in N being dependent only on the et fhn r ig not on fhn 2 r ig. In view of the third line in Eq. (17), the variance in the total number of photon i the um of the variance of the occupation number for all the ingle-particle tate. It i ueful to define the relative root-mean-quare (rm) fluctuation f for tate, the correponding rm fluctuation f for the total number of photon. Thee are, repectively, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hn 2 f i hn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 1 þhn i ¼ > 1; (18) hn i hn i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f : (19) hni Although f > 1, no uch property emerge for f, in fact a we hall ee typically, f 1. III. FLUCTUATIONS IN THE NUMBER OF PHOTONS To calculate the average number of photon it variance, I firt ue the gr canonical enemble with z ¼ 1 (i.e., l ¼ 0). I combine Eq. (13), (15), (17) convert the um over microtate to integral following a tard technique 3 for an aumed three-dimenional container of volume V. Note that there i no Boe condenation for a photon ga, 13 o it i not neceary to plit off a term from the integral, which i neceary for a material ideal ga of boon in a three-dimenional box. Uing the abbreviation x kt A 8pV ð kt Þ3 ðhcþ 3 ; (20) 363 Am. J. Phy., Vol. 83, No. 4, April 2015 Harvey S. Leff 363 Thi article i copyrighted a indicated in the article. Reue of AAPT content i ubject to the term at: Downloaded to IP:

4 the average number of photon i hni ¼ X e ð =kt 1 1 e! A x 2 e x dx =kt 0 1 e x ¼ A½2fðÞ 3 Š ¼ ð2: m 3 K 3 ÞVT 3 : (21) Similarly, converting the um to an integral in Eq. (17) uing Eq. (21) lead to ¼ X! A e =kt 1 e þ e 2=kT =kt ð1 e =ktþ 2 ð 1 0 x 2 e x ð1 e x Þ 2 dx ¼ 1 3 Ap2 ¼ ð2: m 3 K 3 ÞVT 3 ¼ 1:369hNi: (22) An alternative procedure i to write " hni ¼ ¼ z¼1 " z¼1 (23) : (24) With thi procedure, I firt aume general z 6¼ 1, take the needed z derivative of ln Z¼ P lnð1 ze =kt Þ hni, then et z ¼ 1, finally, convert the um to integral. Thi lead, once again, to Eq. (21) (22). Clearly, the average variance given by Eq. (21) (22) both exit for finite T V are of the ame order of magnitude. Uing Eq. (21) (22) in Eq. (19), the relative rm fluctuation in N i f ¼ 2: m 3=2 K 3=2 pffiffiffiffiffiffiffiffi ¼ p 1:170 ffiffiffiffiffiffiffi : (25) VT 3 hni Equation (22) (25) how that i proportional p to hni, the relative fluctuation f i proportional to 1= ffiffiffiffiffiffiffi hni. Thee ame propertie hold for material gae that atify the fluctuation-compreibility theorem, Eq. (8). For example, applying Eq. (8) to air, treated a a claical ideal ga with j T ¼ 1=P; ðkt=vþj T ¼ 1=hNi air, one find hn 2 i air air ¼hNi air, or f pffiffiffiffiffiffiffiffiffiffiffi air ¼ 1= hni air. More generally, Eq. (8) can be written a ¼hNi intenive thermodynamic variable. Returning to the ideal ga, to gain a ene of what thi mean numerically, conider a room with dimenion 3 m 4m 2:5 m thu V ¼ 60 m 3. At a typical room temperature of 300 K, the number denity of photon i hni=v ¼ 5: m 3 the total photon number i hni ¼3: In contrat, the number denity of air molecule at the ame temperature atmopheric preure i hni air =V ¼ 2: m 3 the total number of molecule i hni air ¼ 1: The relative rm fluctuation for photon air, repectively, are f ¼ 5: f air ¼ 2: The average number of air molecule exceed that for photon by eleven order of magnitude, therefore the relative fluctuation for air i much maller. The main point i that the variance exit for the photon ga, for any finite temperature T the relative fluctuation f vanihe in the thermodynamic limit V!1. IV. INAPPLICABILITY OF FLUCTUATION- COMPRESSIBILITY THEOREM Given that the fluctuation in photon number exit, but the iothermal compreibility doe not, it i clear that the fluctuation-compreibility theorem, Eq. (8), fail for the photon ga. To undert why, I outline a proof of the fluctuation-compreibility theorem, modeled after Pathria proof for material particle (not photon). 3 Conider a macrocopic, open ub-volume of material ga particle embedded within a larger ga. Particle can freely flow into out of thi volume; i.e., N i variable. Becaue z ¼ e l=kt i a variable, ktð@=@lþ ¼ zð@=@zþ, thu the variance expreion in Eq. (24) can be written a : (26) The remainder of the proof proceed by auming that V i fixed, but the volume per particle v V=hNi i variable. The right ide of Eq. (26) can be written a ¼ kt v ¼ kt ¼ kt : (27) Finally, a more ueful expreion for ð=@lþ can be obtained uing the Gibb-Duhem equation, dl ¼ v dp dt; (28) where v are the volume entropy per particle. It follow from Eq. (28) that ð=@lþ ¼ v 1, therefore Eq. (26) (27) lead to the fluctuation-compreibility theorem: ¼ kt V 1 ¼ kt v V j T : Thi complete the proof, which I emphaize hold for a material ga. However, for a photon ga, the proof above fail. Equation (26) mut be evaluated at l ¼ 0 thu ha no remaining l dependence. Thu, the tep in Eq. (27) that were ued for material particle cannot be executed. Further, in the Gibb- Duhem equation (28), v P are independent of l, o the expreion, ð=@lþ ¼ v 1 that wa ueful for the material ga doe not hold. In fact, for the photon ga P i not a function of l Eq. (28) reduce to dp/dt ¼ /v ¼ S/V. The latter equation i conitent with the reult obtained by differentiating Eq. (2) comparing the reult with Eq. (3) but i of no help with the proof being attempted. 364 Am. J. Phy., Vol. 83, No. 4, April 2015 Harvey S. Leff 364 Thi article i copyrighted a indicated in the article. Reue of AAPT content i ubject to the term at: Downloaded to IP:

5 Other proof 4,5 of Eq. (8) fail imilarly for the photon ga, the concluion i that the tard proof cannot be ued for the photon ga. Moreover, there cannot exit any other proof becaue, a hown explicitly in Sec. III, for the photon ga the variance of N definitely doe exit, a hown in Sec. I, the iothermal compreibility j T doe not exit. Clearly, Eq. (8) doe not hold for the photon ga. V. A GEDANKEN EXPERIMENT The non-exitence of the iothermal compreibility can be undertood at leat in part by enviaging a Gedanken experiment where the photon ga i contained within a vertical cylinder in a gravitational field. A floating (frictionle) piton i the container ceiling, the wall are maintained at temperature T. Begin with the piton fixed uch that the container volume i V. The number of photon hni adjut, the equilibrium preure P(T) i etablihed in accordance with Eq. (2). If the piton i releaed o that it can float, thermal equilibrium exit only if the piton weight provide an external preure equal to P(T). To meaure the compreibility, add an arbitrarily light grain of to the piton. One might hope to calculate an approximate value of the compreibility uing j T V 1 DV=DP. However, with the wall at fixed temperature, the equilibrium preure of the photon ga doe not change the extra grain caue the piton to drop precipitouly to the container floor; i.e., the untable photon ga collape to zero volume. During the collape, the photon ga follow an irreverible path through non-equilibrium tate. Once equilibrium i re-etablihed, there are zero photon in a zero-volume container. Thu, an arbitrarily mall change in P doe not lead to a correpondingly mall DV there i no way to approximate the iothermal compreibility. The fact that a meaurement of the iothermal compreibility j T i not poible i conitent with the non-exitence of j T etablihed on theoretical ground in Sec. I. VI. CONCLUDING REMARKS Becaue all matter radiate, photon are ubiquitou in the univere. The oldet photon, thoe in the comic microwave background radiation, go back to the big bang. In thi repect, photon are indeed pecial. The photon ga i pecial too in that it i a relatively imple quantum mechanical, relativitic, thermal model, a evidenced by the occurrence of the fundamental contant h, c, k in Eq. (1) (6). Becaue photon can be, are, continually created annihilated by matter, their total number in a cloed box fluctuate continually. Thoe fluctuation are finite, the uggetion that the variance of hni doe not exit becaue the iothermal compreibility doe not exit i incorrect. I have hown here that pecific aumption ued for a material ga to prove the fluctuation-compreibility theorem, namely the proportionality of the variance of hni with the iothermal compreibility, do not hold for photon. ACKNOWLEDGMENTS The author grateful to Raj Pathria Paul Beale for their helpful comment on a firt draft of thi article. a) Electronic mail: hleff@cpp.edu b) Preent addre: SE River Road, Portl, Oregon A detailed dicuion ummary of the thermodynamic propertie of a photon ga can be found in H. S. Leff, Teaching the photon ga in introductory phyic, Am. J. Phy. 70, (2002) reference therein. 2 Becaue P i olely a function of T for the photon ga, ð=@vþ T ¼ 0. Given thi, it i tempting to conclude that ð@v=þ T i infinitely large, but thi i incorrect. The reaon i that the identity ð@v=þ T ¼ 1=ð=@VÞ T doe not hold when the left ide i zero. Rather, it i correct to ay that ð@v=þ T ( thu j T ) doe not exit. 3 R. K. Pathria, Statitical Mechanic, 2nd ed. (Butterworth-Heinemann, Oxford, 1996), pp R. Kubo, Statitical Mechanic (North Holl-Intercience-Wiley, New York, 1965), pp K. Huang, Statitical Mechanic, 2nd ed. (Wiley, New York, 1986), pp For example, ee Ref. 3, p. 172, where after demontrating that the average number of photon i proportional to VT 3, it i written that the latter reult cannot be taken at it face value becaue in the preent problem the magnitude of the fluctuation in the variable N, which i determined by the quantity ð=@vþ 1, i infinitely large. 7 It i worth pointing out that fluctuation in energy are well defined for a photon ga: the variance in energy i kt 2 C v, C v / VT 3 (ee, e.g., Ref. 3, p. 101). Thi ugget that fluctuation in photon number alo exit, contrary to what one might expect from Eq. (7) (8). Thi dichotomy provide further incentive to clarify that the fluctuation in N do indeed exit. 8 An anonymou reviewer of thi manucript ha kindly informed me that a calculation of the fluctuation in N for the photon ga i preented in C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photon Atom: Introduction to Quantum Electrodynamic (Wiley, New York, 1989), pp Thermal photon are known to have zero chemical potential. See, for example, R. Baierlein, The eluive chemical potential, Am. J. Phy. 69, (2001). Two compelling way to argue that l ¼ 0 for thermal photon are: (i) the empirically oberved ditribution of photon frequencie for blackbody radiation agree with the prediction for a quantum ideal boe ga of photon only if the chemical potential l i et equal to zero; (ii) uing Eq. (3) (4) to obtain S(U,V,N) ¼ (4=3)b 1=4 V 1=4 U 3=4, application of the thermodynamic identity l ¼ Tð@S=@NÞ U;V then give l ¼ A jutification ometime given for l ¼ 0 i that the photon number i not fixed, but, rather, i indefinite. See, for example, F. Herrmann P. W urfel, Light with nonzero chemical potential, Am. J. Phy. 73, (2005). Thee author oberve that the indefinite number of photon alone i not ufficient to conclude that l ¼ 0 becaue particle number are not conerved in chemical reaction, where the material contituent have nonzero chemical potential. Indeed, the zero chemical potential reult hold only for thermal photon. For example, if light i in chemical equilibrium with the excitation of matter whoe chemical potential i nonzero e.g., the electron-hole pair in a light emitting diode then the chemical potential of the light mut be nonzero too. The full argument, which entail recognition that l electron þ l hole ¼ l, i given by Herrmann W urfel. See alo P. W urfel, The chemical potential of radiation, J. Phy. C Solid State Phy. 15, (1982). 11 F. Reif, Fundamental of Statitical Thermal Phyic (McGraw-Hill, New York, 1965), Sec Thi quantum tatitic formula for Boe-Eintein ingle-particle tate occupation number {n } can alo be found uing a counting argument. See, for example, D. ter Haar, Element of Statitical Mechanic (Holt, Rinehart Winton, New York, 1964), Sec See, e.g., J. Honerkamp, Statitical Phyic: An Advanced Approach with Application (Springer, Berlin, 1998), pp To ee why, note that the ingle-photon energy q i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ hc=v 1=3, where V ¼ L 3 for an aumed cubical volume, with 2 x þ 2 y þ 2 z, where x, y, z run over p the poitive integer. The ground tate g ha x ¼ y ¼ z ¼ 1 g ¼ ffiffi 3. Uing the reult that the average total number of photon hni /V, it follow that the ratio hn g i=hni /V 2=3! 0 in the thermodynamic limit V!1; i.e., the fraction of photon in the ground (or any other ingle) tate i zero. Note: Thi argument require that we et l ¼ 0 before taking the thermodynamic limit, which i the correct order. If we were to take (incorrectly) the thermodynamic limit firt, we would mitakenly dicover an actually nonexitent ingularity for the ground (or any other) tate. 365 Am. J. Phy., Vol. 83, No. 4, April 2015 Harvey S. Leff 365 Thi article i copyrighted a indicated in the article. Reue of AAPT content i ubject to the term at: Downloaded to IP: