arXiv:1103.3161v1 [physics.gen-ph] 16 Mar 2011 e one u httemdnmcfrua o h htngsc cha two gas hold photon formulas the these for since and and methods formulas gas thermodynamic thermodynamic ideal by that classical pletely out the for pointed formulas Leff thermodynamic basic tween rv httecmlt unu hroyaiso h lc bod black the of distributio thermodynamics Bose-Einstein quantum or complete Planck’s the the that of prove use explicit (by statistics OPEEQATMTEMDNMC OF THERMODYNAMICS QUANTUM COMPLETE ewrs htngstemdnmc,bakbd,bakhl ent hole black body, black thermodynamics, gas photon keywords: 05.70.-a 03.65.Ta, number: PACS el 1 n e 2 eosrtdaddsusdfra n concep and formal discussed and demonstrated [2] Leff and [1] Kelly b hs cuaevle a eotie xlsvl yacrt me accurate by exclusively obtained be can values accurate whose , e one u httemdnmcfrua o h htng h photon formulas the these since for an rameters, methods formulas gas thermodynamic thermodynamic ideal by that completely classical out the pointed for Leff formulas thermodynamic basic oypoo a nrp n eesenHwigbakhl e hole black Bekenstein-Hawking concept and Brogl entropy discuss (de gas we theory photon Finally, atomic body quantum postulate). Bohr quantization degree, the Bohr some in in condition is, bility that meth condition o these stability Planck’s of thermodynamic the Corner-stone re of (with methods. use thermodynamic exactly any quantum practically (without and non-statistical by simply cent), very (ph obtained variables be corresponding (non-statistic can and thermodynamic parameters simple, mentioned by done both or thermodynami be quantum Planck’s can complete the gas the of that ton prove use we explicit work this (by In statistics quantum the of el n e eosrtdaddsusdfra n concept and formal discussed and demonstrated Leff and Kelly eateto hsc,Fclyo cecs 10 oiSad, Novi 21000 Sciences, of Faculty Physics, of Department r H LC OYPOO GAS PHOTON BODY BLACK THE and b hs cuaevle a eotie xlsvl yaccur by exclusively obtained be can values accurate whose , r oieaOrdv´a4 Serbia, Obradovi´ca 4, Dositeja Trg ldnPnoic ak .Kapor V. Pankovi´c, Darko Vladan [email protected] Abstract 1 a iiaiisbtenblack between similarities ual oeEnti distribution), Bose-Einstein r eysmlrt unu sta- quantum to similar very tn ubradpressure) and number otons l ehd.W rv that prove We methods. al) oeEnti distribution). Bose-Einstein aieerraotfwper- few about error lative lc oypoo gas. photon body black d ntropy. l w hrceitcpa- characteristic two old d ersnsaquantum a represents ods esitrrtto fthe of interpretation ie’s so h lc oypho- body black the of cs a iiaiisbetween similarities ual scno ededuced be cannot as htngscnb done be can gas photon y ) nti okw shall we work this In n). ropy atrsi parameters, racteristic no eddcdcom- deduced be annot lc oypoo gas. photon body black hd ftequantum the of thods ulsmlrte be- similarities tual t methods ate r by simple, thermodynamic (non-statistical) methods. Precisely, we shall prove that both men- tioned parameters and corresponding variables (photon number and pressure) can be obtained very simply and practically exactly (with relative error about few percent), by non-statistical (without any use of the Planck’s or Bose-Einstein distribution), quantum thermodynamic meth- ods. Corner-stone of these methods represents a quantum thermodynamic stability condition which is, in some degree, very similar to quantum stability condition in the Bohr quantum atomic theory (de Broglie’s interpretation of the Bohr quantization postulate). Finally, we dis- cuss conceptual similarities between black body photon gas entropy and Bekenstein-Hawking black hole entropy. As it is well-known [3]-[5] quantum statistics of the black body photon gas predicts accu- rately, using Planck’s or Bose-Einstein distribution, the following expressions for black body photon numbers - N, internal energy - U, pressure - p and entropy - S
16πk3ζ(3) N = T 3V = rV T 3 (1) h3c3 8π5k4 U = T 4V N = bV T 4 =2.7NkT (2) 15h3c3 1 U 8π5k4 1 NkT p = = T 4 = bT 4 =0.9 (3) 3 V 45h3c3 3 V 4 U 32π5k4 4 S = = T 3V = bV T 3 =3.6Nk. (4) 3 T 45h3c3 3 Here k represents the Boltzmann constant, h - Planck constant, c - speed of light, T - black body temperature, V - black body volume and ζ(3) ≃ 1.202 - Riemann zeta function, while
k r = 60.4( )3 (5) hc 8π5k4 b = (6) 15h3c3 represent parameters that can not be obtained by relatively inaccurate thermodynamic methods but that can be obtained exactly by accurate methods of the quantum statistics (with explicit use of the Planck’s or Bose-Einstein distribution). However, suppose that photon gas is captured in a hollow, spherical black body with rela- tively large radius R at temperature T . Suppose that, in the thermodynamic equilibrium, photons within hollow, spherical black body propagate, practically, over black body internal surface. Precisely, suppose that these photons propagate over a black body great internal circle with radius R and circumference 2πR, according to the following quantum thermodynamic stability condition h
2 Expression (7) implies
h 2πR =
kT <ν>= (10) h so that hc <λ>= (11) kT and R 2πRkT
3 or, according to the previously introduced expressions (5), (10), (11) ,
k k
k 3 Now, it can be observed that term 59.2( hc) in (16) is practically (with relative error about 2%) identical to parameter r (5). It admits statement that
k r =6π2( )3 (17) hc represents correctly, simply quantum thermodynamically, non-statistically (without any use of the Planck’s or Bose-Einstein distribution) derived value of the parameter r. Then, right hand of (16) is practically identical to right hand of (5) so that we can suppose that left hand of (16) is practically identical to left hand of (5) too, i.e.
N =
In this way we reproduced correctly, practically accurately (with relative error about 2 Further, it can be simply quantum thermodynamically, non-statistically (without any use of the Planck’s or Bose-Einstein distribution), supposed that power, P , of the black body radiation per unit area, A, can be approximated by the following expression P
π3 =1.033 ≃ 1 (20) 30 it follows 3 P ≃ π c 4 ≡ 4 <ν> kT = bT σT (21) 2 A 30 <λR> 4 representing Stefan-Boltzmann law derived simply thermodynamically, non-statistically (with- out any use of the Planck’s or Bose-Einstein statistics), where, of course, σ represents the Stefan-Boltzmann constant. It can be observed that this expression is very close (with rela- tive error about 3%) to the exactly, quantum statistically obtained Stefan-Boltzmann law (by explicit use of the Planck’s or Bose-Einstein distribution). Expression (19) implies the following simple quantum thermodynamic, non-statistical (with- out any use of the Planck’s or Bose-Einstein distribution), expression for the pressure of the photon radiation emitted by black body
4 It, according to (10), (11), (16), can be simply transformed in the following way
kT 4 k 2 NkT p =4π2 = 6π2( )3T 3kT = . (23) em <λ>3 6 hc 3 V On the other hand, according to (9)-(11), simple quantum thermodynamic, non-statistical (without any use of the Planck’s or Bose-Einstein distribution), expression for the pressure of the reflected photon radiation in a hollow spherical black body equals approximately
mc2 R 1 NkT 1 NkT pref = N 2 = 4 3 = . (24) 4πR 3 3 πR 3 V
Then, total pressure of the black body photon gas obtained by simple quantum thermody- namic, non-statistical (without any use of the Planck’s or Bose-Einstein distribution) methods equals NkT p = p + p = . (25) em ref V This expression is very close (with relative error about 10 It can be added that when photon gas pressure is determined by simple quantum thermody- namic, non-statistical (without any use of the Planck’s or Bose-Einstein distribution) methods (25), parameter b, according to (3), can be determined by these methods too. In this way it can be concluded that complete quantum thermodynamics of the black body photon gas can be deduced by simple, quantum thermodynamic, non-statistical (without any use of the Planck’s or Bose-Einstein distribution) methods, even if, of course, complete quantum thermodynamics of the photon gas represents an approximate theory of the accurate quantum statistical theory of the photon gas (with explicit use of the Planck’s or Bose-Einstein distri- bution) . Consider, finally entropy of the black body photon gas according to simple, quantum ther- modynamic, non-statistical (without any use of the Planck’s or Bose-Einstein distribution) methods. It, in fact, represents introduction of (25) in (4) which, according to (16), yields
2 kT 3 3 V ≃ V S =4kN = k24π ( ) V = k 3 k 3 . (26) hc π <λR > <λR > Accurate quantum statistical expression for the black body photon gas entropy (4) can be transformed in 5 4 2 32π k 3 4π V ≃ V S =3.6Nk = 3 3 T V = k( ) 3 0.88k 3 . (27) 45h c 45 <λR > <λR > In both cases, i.e. for (26), (27), entropy of the black body photon gas is practically equivalent to the quotient of the black body volume V and ”minimal quantum thermodynamic volume” 3 <λR > multiplied by Boltzmann constant k. It represents very interesting result. All this is similar to the Bekenstein formula for the black hole entropy
ABH ABH SBH = k 2 = k 2 . (28) (2LP ) (2λRP )
5 Here ABH represents the surface of the black hole which, for a spherical, Schwarzschild black 2 2 hole with horizon radius RS, equals ABH = 4πRS. Also, here (2LP ) represents ”elementary ¯ 1 hc 2 surface of the black hole” since LP = ( G ) = λRP represents the Planck length equivalent to h the reduced Compton wave length of a particle with Planck mass, whereh ¯ = 2π represents the reduced Planck constant and G - Newtonian gravitational constant. As it is well-known this Planck length represents the minimal length with the physical meaning in the quantum field theories. Since entropy represents an additive variable, expression (26) we can transform in the fol- lowing way
≃ V ≃ Vshell Vint ≡ A S k 3 k 3 + k 3 Sshell + Sint = k 2 + Sint. (29) <λR > <λR > <λR > <λR > Here Vshell A<λR > A Sshell = k 3 = k 3 = k 2 (30) <λR > <λR > <λR > can be considered as the entropy in a tiny (with width < λR >) spherical shell nearly black 2 body surface A = 4πR , while Sint = S − Sshell represents the entropy of the rest, internal (under shell) volume of the black body photon gas. It can be observed that Bekenstein black hole entropy (28) has practically identical form as the entropy of the black body photon gas spherical shell (30). It is not any surprise since black hole represents, at it has been proved by Bekenstein, Hawking and other [6], [7], an especial form of the black body. Classically speaking, according to the cosmic censorship conjuncture, except information on the black hole total mass-energy (for a Schwarzschild black hole), there is no other information on the black hole physical characteristics inside horizon. Quantum field theory, however, roughly speaking, according to the possibility of the particle-antiparticle pair creation nearly horizon, admits additional considerations of the events that occur in 2λRP a tiny spherical shell (with width about 2λRP ) nearly horizon corresponding to the Bekenstein- Hawking black hole entropy as, practically, black body shell entropy. On the other hand, according to the cosmic censorship conjuncture, black hole does not hold internal (under shell) entropy in difference to the other black bodies for which cosmic censorship conjuncture is not satisfied. In conclusion we can shortly repeat and point out the following. Kelly and Leff demonstrated and discussed formal and conceptual similarities between basic thermodynamic formulas for the classical ideal gas and black body photon gas. Leff pointed out that thermodynamic formu- las for the photon gas cannot be deduced completely by thermodynamic methods since these formulas hold two characteristic parameters, r and b, whose accurate values can be obtained exclusively by accurate methods of the quantum statistics (by explicit use of the Planck’s or Bose-Einstein distribution). In this work we prove that the complete quantum thermodynamics of the black body photon gas can be done by simple, thermodynamic (non-statistical) meth- ods. We prove that both mentioned parameters and corresponding variables (photon number and pressure) can be obtained very simply and practically exactly (with relative error about few percent), by non-statistical (without any use of the Planck’s or Bose-Einstein distribu- tion), quantum thermodynamic methods. Corner-stone of these methods represents a quantum thermodynamic stability condition that is, in some degree, very similar to quantum stability
6 condition in the Bohr quantum atomic theory (de Broglie’s interpretation of the Bohr quanti- zation postulate). Finally, we discuss conceptual similarities between black body photon gas entropy and Bekenstein-Hawking black hole entropy.
References
[1 ] R. Kelly, Am. J. Phys., 49 (1981) 714
[2 ] H. D. Leff, Am. J. Phys. 70 (2002) 792
[3 ] D. J. Amit, J. Werbin, Statistical Physics: An Introductory Course (World scientific, Singapore, 1995)
[4 ] L. D. Landau, E. M. Lifshitz, Statistical Physics (Adison-Wesley, Reading, MA, 1958)
[5 ] A. Isihara, Statistical Physics (Academic Press, New York, 1971)
[6 ] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermody- namics (University of Chicago Press, Chicago, 1994)
[7 ] R. M. Wald, The Thermodynamics of Black Holes, gr-qc/9912119
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