On Irreversible Radiation Processes

Transcription

1 On Irreversible Radiation Processes Huaiyu Zhu SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 Submitted to Physical Review Letters, Feb 1997 On Irreversible Radiation Processes Huaiyu Zhu Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (March 11, 1997) Detailed calculation of the irreversible radiation processes leading to thermal radiation demonstrates that entropy is produced locally at each absorption and emission of a photon, lending strong support to an irreversible quantum theory d (q.sm), j (uct), Ln (noneq.td), 1.0.-m (qed) Based on some recent results in statistics [1], it was shown that the standard quantum theory is mathematically incompatible with the second law of thermodynamics, and an alternative irreversible quantum theory was formulated []. In this letter we examine in detail the special case of thermal radiation and show that Planck's original invention of the quantum as an explanation of entropy and irreversibility [3{5] is completely vindicated down to the most microscopic level: Entropy is produced at each emission and absorption of a single photon. This appears to be compatible with the irreversible quantum theory proposed. I. KNOWN RESULTS as Planck's law for the spectral distribution of energy of thermal radiation may be written u = g "(e "=T ; 1) ;1 (1) where " = h and g = 8 c 3 () is the number of photon eigenstates per unit volume per unit frequency [6]. Hereafter we shall omit explicit references to \per unit volume". We assume that the temperature unit is so chosen that the Boltzmann constant k =1. To obtain formulas in the SI units one only needs to replace temperature T by kt and entropy s by s=k. Let N M be the number of photons in the state M with frequency M, Planck's law can be simplied to N M =(r ;1 M ; 1) ;1 r M = e ;"=T : (3) zhuh@santafe.edu. Fax:

3 We shall often omit explicit references to M. According to modern theory it is understood that this average photon number N arises from the Bose-Einstein distribution p N =(1; r)r N = r N =z (4) where z = P N r N =(1; r) ;1 is the partition function. The entropy of distribution p is s = ; N p N log p N = ; r log r ; log(1 ; r) (5) 1 ; r which exactly corresponds to Planck's law of spectral distribution of entropy [3] s = g 1 1 ; r log 1 1 ; r ; r 1 ; r log r 1 ; r = g s: (6) The uctuation of the photon numbers N = (N ; N), hence also uctuation of total photon energy u =(g "N), is given by Einstein as N = N p N (N ; N) = r 1 ; r + r 1 ; r : (7) The rst term arises from pure particle phenomena while the second term arises from pure wave phenomena [7,4,8]. With hindsight we can say that adoption of this uctuation formula is exactly the essential \interpolation step" in Planck's October 1900 paper, while its probabilistic interpretation is the essential step of his December 1900 paper. Denote = 1 T = ds du : (8) Using highly retrospective notations and ignoring historical order, the early conceptual development of wave-particle duality may be summarized as Wien: u = g " e " =) ; du d = "u Rayleigh: u = g Planck: u = g " e " ; 1 Einstein: u = ; du d : =) ; du d = u g (= ; du d = "u + u g (9) The precise historical account can be found in the excellent studies [4,8,5]. II. RADIATION PROCESS According to Planck, this distribution is an equilibrium resulting from irreversible radiation processes, ie. the absorption and emission of radiation by some \virtual oscillators". Although it is now known that these oscillators are in fact bound electrons, in the following

4 we shall treat them as \quasi-classical particles" which cannot be created or annihilated but may jump among energy eigenstates denoted by " n, etc. Later we shall discuss the eect of assuming them to be fermions, but our immediate concern is to examine the details of such irreversible processes. As was known since Kirchho [3], the exact properties of these oscillators will have no eect on the equilibrium distribution of the radiation. This point will also become obvious in our considerations. Suppose " = " m ; " n > 0. Denote by q n the probability of an oscillator being in state n. According to Einstein [9], the time rates of photon emission and absorption are with Emission: q m (B n mu + A n m) (10) Absorption: q n B m n u (11) A n m = g "B n m B n m = B m n : (1) We do not need Einstein's assumption that the oscillators are in a Boltzmann distribution It will come out naturally. Expressed in photon numbers, Einstein's mechanism may be equivalently written as Emission: q m A(N +1) (13) Absorption: q n AN: (14) According to Bothe [10,4], this relation involving average photon numbers could be explained by the following relation involving actual photon numbers Emission: q m A(N +1)p N (15) Absorption: q n A(N +1)p N +1 : (16) In the sequel we shall denote C = A(N + 1). The symmetry between (15) and (16) represents the \reciprocity" or \detailed balance", which is often mistaken as representing \microscopic reversibility" [6]. The former concerns the symmetry between two states in the forward dynamics P (x t+dt = mjx t = n) =P (x t+dt = njx t = m) (17) which originates from Hermitian operators, while the latter concerns the symmetry between forward and backward dynamics P (x t+dt = mjx t = n) =P (x t = mjx t+dt = n) (18) which originates from unitary operators. Their distinction can be seen clearly in the simple example of the standard discrete-time random walk: At any time t the probabilityofmoving from location x to x +1is exactly the same as moving from x +1to x, yet the process is patently irreversible. 3

5 The Einstein-Bothe mechanism implies that, if there is ever to be an equilibrium, it should hold that p N +1 p N = q m q n (19) for any combinations of n m M satisfying " m ; " n = h M. It is therefore clear that the above quotient can only be of the form e ;("m;"n)=t, where T does not depend on n m M. This immediately leads to the Boltzmann distribution for the oscillators and to the Bose- Einstein distribution for the photons, without any extra hypothesis. It remains to be shown that the Einstein-Bothe mechanism is indeed an irreversible process which converges to this unique equilibrium. III. ENERGY, ENTROPY AND TEMPERATURE For purely notational convenience we shall call the oscillators as electrons. The photon and electron entropies and their changes are s := ; N ds := ; N p N log p N s e := ; P n q n log q n : (0) dp N log p N ds e := ; P n dq n log q n : (1) In the \microscopic system" involving only m n N N +1, the time rates of entropy changes are given by Their sum is Emission: 1 ds C dt q m p N log p N p N +1 Absorption: q n p N +1 log p N +1 p N 1 ds e C dt q m p N log q m q n q n p N +1 log q n q m () ds dt = C(q mp N ; q n p N +1 )log p Nq m p N +1 q n 0: (3) That is, the total entropy of this micro-system increases with time, whenever the equilibrium condition (19) is not satised. Summing over all the micro-systems m n N N +1 shows that the whole process is irreversible. This summing up is allowed if the transition between states m n with the change of photon number between N and N +1 is independent of probabilities involving other numbers. Such independence assumptions are invalid in any reversible theory. The entropy production can in fact be localized even further. Let us dene the photon and electron temperatures associated with these microscopic distributions T := " log(p N =p N +1 ) T e := 4 " log(q n =q m ) : (4)

6 The time rate of energy changes are given by 1 dq 1 dq e Emission: C dt q m p N " C dt ;q m p N " Absorption: ;q n p N +1 " q n p N +1 ": (5) Therefore, ds ; dq T =0 ds e ; dq e T e =0: (6) dq T + dq e T e = ds 0 for dt > 0: (7) This is interpreted as that neither photons nor electrons produce entropy by themselves, but that entropy is produced by the exchange of energy between subsystems at dierent temperatures. Furthermore, because of energy conservation, dq + dq e = 0, the last inequality above is equivalent to dq dt 0 () dq e dt 0 () T T e (8) which means energy always ows from high temperature to low temperature. This law is irreversible because dt is antisymmetric to time reversal. Therefore we have localized the entropy production at the microscopic level to the emission and absorption of a single photon. At equilibrium the temperatures of all the dierent micro-systems are equal to each other, so the entropy production ceases even though energy continues to ow back and forth between photons and electrons. In the above wehave treated the oscillators, although called electrons, as quasi-classical particles. It is clear that analogous results will hold for fermions: The only dierence is that the equilibrium distribution for the fermions will be Fermi-Dirac distribution The distribution of the photons will remain the same. The distribution of one subsystem at equilibrium is insensitive tothedynamics of the other subsystems it is in equilibrium with, because the equilibrium is simply characterized by equations like (19) and the temperature is given by equations like (4). Generally, the equilibrium distribution for an arbitrarily complicated system involving dierent kinds of particles may beobtainedby maximizing its entropy, constrained by applicable conservation and symmetry laws and other restrictions. This is the reason for the importance of such constraints in physics. On the other hand, physicists often nd conservation laws by looking at various equilibrium macrostates, with the faith that \anything allowed in nature will happen in nature". Obviously this faith could only be justied if fundamental physical processes are irreversible. For it is well known that, due to Liouville's theorem, the ergodicity hypothesis could not hold for reversible processes [5,6]. Needless to say, most of the key ideas adopted here can be at least traced back to to Planck, including the idea that thermal radiation is the equilibrium resulting from an 5

7 irreversible radiation process, that each component of radiation possesses its own entropy and temperature, and that equilibrium is characterized by equal temperature among all the components. The new contributions here are the actual calculations which vindicates Planck's original intention, and the demonstration that these ideas can be pushed to the most microscopic level. IV. QUANTUM THEORY We must now reconcile this irreversible mechanism with the fact that the standard quantum theory is reversible. It is generally accepted that a theory which \explains" radiation must lead to Einstein's coecients of absorption and emission, and to Einstein's energy uctuation formula in equilibrium. Since the latter involves a term for particle interaction, it cannot be derived by the standard quantum theory which is essentially awave theory. A careful examination of the derivations by Born, Heisenberg and Jordan [11] reveals that a crucial additional assumption incompatible with the theory itself is assumed. This is the ergodicity assumption which claims that the time average is equal to the ensemble average, without which only the wave uctuation term could be obtained. The treatment of Einstein's coecients is even more obvious: Dirac [1] essentially treated the process as independent random jumps. Examination of other phenomena reveals that this is always the intended interpretation in practice. For example, in Born's probabilistic interpretation each matrix element is converted to transitional probability independent of the others. It is also quite clear that Heisenberg regarded the random jumps to occur at any possibility of measurement [13], ie., any interaction. Therefore an irreversible quantum theory is needed to account for the actual interpretations and applications. Suppose the state of a free photon gas is given in the occupation number representation as. That is, we assume the free particle propagator has been transformed away except for a phase factor. Let V (t) denote the interaction Hamiltonian in this representation. In the standard theory, the transition probability from initial state i to nal state f is given by Dyson's formula P fi = j fs i j = S = k k m 1 ::: m k;1 Z Z m 1 i t 1 <:::<t k V t k fm k;1 V t 1 (9) S k = V (t k ) V (t 1 ) (30) k t 1 <:::<t k Vba t = b V (t) a (31) where m 1 ::: m k are intermediate states, and the corresponding sums are over an orthonormal basis. That is, P m m m is the identity operator. This reversible dynamics is of course incompatible with the concept of equilibrium. By equating his derived transition probability with Einstein's coecients, Dirac eectively adopted the following formula 6

8 P fi = k P k fi k m 1 ::: m k;1 1 Q 1 ba = t!1 lim t Z t k Q 1 fm Q1 k;1 m 1 i k! (3) Vba t t (33) where Q 1 ba is proportional to the transition rate from a to b in a single transition. The \occupation number representation" is used as the orthonormal basis. It is extremely importantto note that now the choice of basis is physically signicant. This formula is only approximate for innite time, and it is not relativistically invariant because time is treated dierently than space. To be correct for macroscopic phenomena, it must be an approximation to a relativistic and irreversible formula. From some general considerations one may propose an intermediate formula, in which time as well as space are in superposition while the interaction patterns are in mixture [], P fi = k P k fi P k fi m 1 ::: m k;1 Z m 1 i t 1 <:::<t k V t k fm k;1 V t 1 : (34) The identity of particles is in superposition by using the occupation number representation. The dierence with the standard theory is that the change of photon numbers is now in mixture. This formula is relativistically invariant and describes an irreversible process. If it corresponds to Dirac's formula, then in the long run (for large t) we should have Some numerical simulations suggest that instead Pfi k tk (k)! P fi k;1 P k fi tk k! : (35) t k (k ; 1)! : (36) Therefore, in the macroscopic approximation the number of interactions satisfy a clustered Poisson distribution. If we regard a pair of interactions as a single event then these events are independent, recovering Dirac's approximation. Further studies are needed to show whether such pairs are actually due to a correlation between emission and absorption which should then be considered as a single scattering. Such a statistical correlation, however, does not change the fact that the distribution will approach equilibrium and that the equilibrium can only be as derived above. This statement would break down only if the re-emitted photon were always in the same state as the incident photon, which appears not to be the case. At this moment it is probably safe to say that the irreversible quantum theory appears to be able to explain certain phenomena both microscopically and macroscopically, in ways not possible by either reversible quantum theory or classical statistical mechanics alone. If the actual interaction Hamiltonians for electrons and photons are used, it will be able to predict the speed a non-equilibrium radiation in cavity approaches equilibrium. Without detailed calculation, it is still possible to arrive at a rough estimate of the equilibration 7

9 time [] which may be experimentally tested. It is also worth looking for other phenomena which converges to equilibrium more slowly while amenable to microscopic treatment. We conclude with a conjecture that in general all quantum processes produce entropy in similar ways, and that the second law can be entirely explained microscopically in quantum phenomena. The general relation between quantum theory and irreversibility is discussed in []. My work in SFI is sponsored by TN,Inc. [1] H. Zhu. On information and suciency. SFI working paper Submitted to Ann. Stat., [] H. Zhu. Quantum theory and irreversibility. Submitted to Phys. Rev. Lett., [3] M. Planck. Treatise on the Theory of Heat Radiation. Blakiston, Philadelphia, Trans. by M. Masius from Vorlesungen uber die Theorie Warmestrahlung (nd ed, 1913). [4] E. Whittaker. A History of the Theories of Aether and Electricity, volume II: Modern Theories 1900{196. Harper, New York, [5] T. S. Kuhn. Black-Body Theory and the Quantum Discontinuity, 1894{191. Oxford Univ., London, [6] W. Yourgrau, A. van der Merwe, and G. Raw. Treatise on Irreversible and Statistical Thermodynamics: An Introduction to Nonclassical Thermodynamics. Dover, New York, 198. [7] A. Einstein. Zum gegenwartigen Stand des Strahlungsproblems. Phys. Zs., 10:185{193, (On the current state of the problem of radiation). [8] M. Jammer. The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York, [9] A. Einstein. Zur Quantentheorie der Strahlung. Phys. Zs., 18:11{18, (On the quantum theory of radiation). [10] W. Bothe. Z. Phys., 0:145, 193. [11] M. Born, W. Heisenberg, and P. Jordan. Zur Quantunmechanik II. Z. Phys., 35:557{615, 195. (On quantum mechanics II). [1] P. A. M. Dirac. The quantum theory of the emission and absorption of radiation. Proc. R. Soc. Lond., A, 114:43{65, 197. [13] W. Heisenberg. The Physical Principles of The Quantum Theory. Dover, New York,