Fluctuation theorem for black-body radiation
FFluctuation theorem for black-body radiation
B. Cleuren and C. Van den Broeck
Hasselt University - B-3590 Diepenbeek, Belgium
The fluctuation theorem is verified for black-body radiation, provided the bunching of photons istaken into account appropriately.
PACS numbers: 02.50.-r,05.40.-a,42.50.Ar,42.50.Lc
I. INTRODUCTION
The study of black-body radiation has played a promi-nent role in the discovery of quantum mechanics. Morerecently, the quantum features of light have been revealedin many beautifull optical experiments, and found to bein full agreement with the quantum theory of electro-magnetic radiation [1, 2, 3]. In particular, we cite thebunching of photons. Bunching refers to the tendency ofphotons to arrive together at closely spaced detectors, re-sulting in super-Poissonian counting statistics (variancelarge than the mean). While a classical explanation isavailable [4], the detailed understanding and descriptionof the phenomenon requires quantum field theory [1, 5].The bunching of photons, and more generally of bosons[6], is usually explained in terms of the more familiarquantum statistical exchange force which introduces aneffective attraction between identical bosons, when theirwavefunctions overlap [7]. The phenomenon of bunchingwas first observed by Hanbury Brown and Twiss [8] asthe correlation between intensities of light from a singlestar falling on different detectors. Bunching of photonsin a single detector experiment was later observed [9]and found to be in agreement with a simplified quantummechanical argument by Glauber [10]. For the case ofequilibrium black-body radiation, the photon count hasbeen calculated explicitly [5] and is given by a negativebinomial distribution rather than the Poissonian distri-bution which describes independent arrivals.In this Letter, we show that this result for the photoncounting statistics is essential to find agreement with arecent result from nonequilibrium statistical mechanics,namely, the so-called fluctuation theorem. This theo-rem states that the probability distribution P (∆ S ) toobserve an entropy production ∆ S during a time inter-val t in a nonequilibrium steady state obeys the follow-ing symmetry relation for asymptotically long values of t [11, 12, 13, 14, 15, 16, 17]: P (∆ S ) P ( − ∆ S ) ∼ exp { ∆ S/k } . (1)In words, the probability of observing a positive entropychange is exponentially larger than that of the corre-sponding negative change. When the system starts ina state of canonical equilibrium which is subsequentlyperturbed by a time-dependent change of the Hamilto-nian, the above fluctuation theorem, referred to as thetransient fluctuation theorem, is valid for all times, and not just asymptotically large ones [16, 18].The fluctuation theorem has been verified in several the-oretical [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]and experimental settings [32, 33, 34, 35, 36, 37, 38]. Thephysical origin of the fluctuation theorem is to be foundin the time-reversal symmetry of the underlying Hamil-tonian dynamics. As such, this result can be viewed as ageneralization of the ideas of Onsager [39].We shall investigate the steady state of radiation ex-change between two black bodies, at equilibrium at differ-ent temperatures, when they are connected to each otherthrough a small aperture. To present the problem andto contrast the role of the quantum mechanical bunchingfor photons with that of classical particle exchange, westart by studying the classical counterpart of effusion ofthe ideal gases. II. EFFUSION OF A CLASSICAL IDEAL GAS
Consider two infinitely large reservoirs A and B , eachcontaining classical ideal gases at (canonical) equilibriumwith temperatures T A and T B and densities ρ A and ρ B ,respectively. We perform the following experiment (seefig. 1). During a fixed time interval t , a small hole ofsurface area σ between the reservoirs is opened. We as-sume, for better comparison with the photon crossings,that the opening contains an energy filter, allowing onlyparticles with kinetic energy in the range E ± δE/ U andnet number of particles ∆ N transferred from A to B aremeasured. We consider the limit of a small energy win-dow, δE (cid:28) E , so that ∆ U = E ∆ N , and the exchangein the number of particles is the only relevant variable.Since the hole is small enough so that the canonical equi-libria in the reservoirs are not significantly perturbed, thecorresponding entropy change ∆ S is given by standardthermodynamics:∆ S = (cid:26)(cid:18) T B − T A (cid:19) E + (cid:18) µ A T A − µ B T B (cid:19)(cid:27) ∆ N. (2)Inserting the well-known espressions [40] for the chemicalpotentials µ α of the ideal gases in reservoirs α ∈ { A, B } ,one finds: µ A T A − µ B T B = k log (cid:32) ρ A ρ B (cid:20) T B T A (cid:21) (cid:33) . (3) a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y T A , ρ A T B , ρ B ∆ U , ∆ Na) b)
FIG. 1: (Color online) Set-up for the effusion of ideal gases:a) Two separate ideal gases in equilibrium at their respec-tive temperature and density. b) During a time t , a smallhole is opened and the net transfer of energy ∆ U and ∆ N ismeasured. Since the particle exchange ∆ N is the result of individ-ual gas particle crossings, it is obviously a random vari-able, and hence so is ∆ S . The probability distribution P (∆ N ) can be calculated as follows. We define p α ( j )as the probability that j particles leave the reservoir α in the specified time interval. As all the particles moveindependently of one another, p α ( j ) is Poissonian [41]: p α ( j ) = ( ν α t ) j j ! e − ν α t . (4)The escape rate ν α can be calculated following standardarguments from kinetic theory (see fig. 2). We get ν α = σρ α √ πmkT α E kT α e − E /kT α δE. (5)Since the escape of particles from the left and right reser-voirs are independent phenomena, P (∆ N ) is given by thesum P (∆ N ) = ∞ (cid:88) i =0 p A (∆ N + i ) p B ( i ) . (6)Combining eqs. (4) and (6) leads to the following explicitexpression for P (∆ N ): P (∆ N ) = e − t ( ν A + ν B ) (cid:18) ν A ν B (cid:19) ∆ N I ∆ N (2 t √ ν A ν B ) . (7)Recalling that the modified Bessel function I ∆ N is aneven function of ∆ N , the fluctuation theorem is verifiedby direct substitution of this result in eq. (1).We mention a simpler proof, more clearly related to theunderlying micro-reversibility. Consider the exchangeprocess, with ∆ N + i particles going from A → B , and i particles from B → A . This occurs with probability p A (∆ N + i ) p B ( i ). The time reversed process, with i par-ticles going from A → B and ∆ N + i particles from B → A , has probability p A ( i ) p B (∆ N + i ). Their ratio θ σ v cos( θ) dt v FIG. 2: The escape rate is determined by counting the parti-cles that have the correct kinetic energy and which are ableto reach the opening in a short time dt (these are located inthe cylinder with volume σv cos( θ ) dt , v = p E /m ). In-serting the Maxwellian velocity distribution and adding up allcontributions from the different angles leads to the expressioneq. (5). satisfies the following detailed fluctuation theorem: p A (∆ N + i ) p B ( i ) p A ( i ) p B (∆ N + i ) = (cid:18) ν A ν B (cid:19) ∆ N = (cid:32) ρ A ρ B (cid:20) T A T B (cid:21) exp (cid:26)(cid:20) T B − T A (cid:21) E k (cid:27)(cid:33) ∆ N = e ∆ Sk , (8)using eqs. (5) and (2). The fluctuation theorem itselffollows immediately. We have P (∆ N ) = ∞ (cid:88) i =0 p A (∆ N + i ) p B ( i )= ∞ (cid:88) i =0 (cid:18) ν A ν B (cid:19) ∆ N p A ( i ) p B (∆ N + i )= (cid:18) ν A ν B (cid:19) ∆ N P ( − ∆ N ) . (9) III. BLACK-BODY RADIATION
We now turn to a similar set-up for the study of fluctu-ations in black-body radiation. Two large empty cavities( A and B ), whose walls are kept at a fixed, but differenttemperatures, T A and T B respectively, act as sources ofblack-body radiation (see fig. 3). During a fixed time in-terval t , radiative exchange becomes possible by openinga small aperture (surface area σ ), which permits free pas-sage of photons with a frequency in the range ω ± δω/ δω (cid:28) ω . Following standard thermodynamics, the en-tropy change ∆ S , upon transfer of a net number of pho-tons ∆ N from A to B (and hence of energy (cid:126) ω ∆ N ), isgiven by: ∆ S = (cid:18) T B − T A (cid:19) (cid:126) ω ∆ N. (10) ∆ U T A T B a) b) FIG. 3: (Color online) Set-up for black-body radiation: a)Two cavities are equilibrium each at its own temperature. b)During a time t , a small hole is opened and the net transferof energy ∆ U is measured. Photons coming from the two different reservoirs are in-dependent of each other. Hence the probability P (∆ N )to observe a net transfer of ∆ N photons from A to B isagain given by eq. (6). However, photons coming fromthe same reservoir are not independent, and their escapeis no longer governed by a Poisson distribution. The cal-culation of p α ( j ), which is also referred to as the photoncounting distribution, requires a fully quantum mechan-ical description of the electromagnetic field [1, 5]. Forlarge times ( tδω (cid:29) p α ( j ) = Γ( j + νt ) j !Γ( νt ) (cid:0) − e − (cid:126) ω /kTα (cid:1) νt (cid:0) e − (cid:126) ω /kTα (cid:1) j , (11)where ν is defined as ν = σω δω (2 πc ) . (12)It is now a matter of simple algebra to verify that thefollowing detailed fluctuation theorem is obeyed: p A (∆ N + i ) p B ( i ) p A ( i ) p B (∆ N + i ) = e “ TB − TA ” (cid:126) ω ∆ N/k . (13)Consequently, following the steps of eq. (9), the fluctua-tion theorem is also verified.We emphasize that the correlations between the pho-tons, resulting in the negative binomial distribution, isan essential ingredient. A semiclassical approach, whichassumes statistical independence between photons, can-not be reconciled with the fluctuation theorem (see Ap-pendix). IV. DISCUSSION
We close with a number of comments on the above re-sults.We have presented the analysis above for the case of amonochromatic window connecting the reservoirs. Thispermits the derivation of a detailed fluctuation theorem.It is also revealing to note that the entropy productionbecomes identically zero for the classical gas for a par-ticular choice of the energy window: the r.h.s. of eq. (8) is identically equal to unity for a particular value of theenergy, namely, E = k T A T B T A − T B ln (cid:40) ρ A ρ B (cid:20) T A T B (cid:21) (cid:41) . (14)At this specific value of the kinetic energy, the one-particle energy distributions in reservoirs A and B crosseach other, so that the particles with this energy havethe same number density on both sides. Such an equilib-rium state can be used to reach optimal thermodynamicCarnot and Curzon-Ahlborn efficiencies [42]. This typeof equilibrium can also be achieved with electrons [43],but not with photons because the latter have zero chem-ical potential.The fluctuation theorem for the effusion of an ideal gasis valid for all times, while for photons the theorem hasbeen proven here only for large times. This is consistentwith the observation that the effusion of an ideal gas is aprocess without memory. In such a case, the distinctionbetween the steady state and transient versions of thefluctuation theorem disappears. A transient theorem forphotons valid for all times can also be obtained in prin-ciple, but this would require the exact evaluation of thetransient photon count, starting from zero and progress-ing to the steady state, after the aperture is opened.As expected from the general argument connectingbunching to the statistical exchange interaction, one findsthat fermions display anti-bunching [44]. While this phe-nomenon has no classical analogue, it is surprising tofind, for example, that the statistics of an electron cur-rent through a small channel can be fully reproduced bya simple classical random walk model, namely, the sym-metric exclusion process [31]. For this simple model, thefluctuation theorem is again verified. The validity of thefluctuation theorem for fermions has also been confirmedby quantum field theoretic calculations [45]. V. APPENDIX
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