The Theory of Heat Radiation (1914) — part 12 of 12

190. Conclusion. — The theory of irreversible radiation proc esses here developed explains how, with an arbitrarily assumed initial state, a stationary state is, in the course of time, established in a, cavity through which radiation passes and which contains oscillators of all kinds of natural vibrations, by the intensities and polarizations of all rays equalizing one another as regards magnitude and direction. But the theory is still incomplete in an important respect. For it deals only with the mutual actions of rays and vibrations of oscillators of the same period. For a definite frequency the increase of entropy in every time element until the maximum value is attained, as demanded by the second principle of thermodynamics, has been proven directly. But, for all frequencies taken together, the maximum thus attained does not yet represent the absolute maximum of the entropy of the system and the corresponding state of radiation does not, in gen eral, represent the absolutely stable equilibrium (compare Sec. 27). For the theory gives no information as to the way in which the intensities of radiation corresponding to different frequencies equalize one another, that is to say, how from any arbitrary initial spectral distribution of energy the normal energy distri bution corresponding to black radiation is, in the course of time, developed. For the oscillators on which the consideration was based influence only the intensities of rays which correspond

CONSERVATION OF ENERGY AND INCREASE OF ENTROPY 215

to their natural vibration, but they are not capable of changing their frequencies, so long as they exert or suffer no other action than emitting or absorbing radiant energy.1

To get an insight into those processes by which the exchange of energy between rays of different frequencies takes place in nature would require also an investigation of the influence which the motion of the oscillators and of the electrons flying back and forth between them exerts on the radiation phenomena. For, if the oscillators and electrons are in motion, there will be impacts between them, and, at every impact, actions must come into play which influence the energy of vibration of the oscillators in a quite different and much more radical way than the simple emis sion and absorption of radiant energy. It is true that the final result of all such impact actions may be anticipated by the aid of the probability considerations discussed in the third section, but to show in detail how and in what time intervals this result is arrived at will be the problem of a future theory. It is certain that, from such a theory, further information may be expected as to the nature of the oscillators which really exist in nature, for the very reason that it must give a closer explanation of the physical significance of the universal elementary quantity of action, a significance which is certainly not second in importance to that of the elementary quantity of electricity.

i Compare P. Ehrenfest, Wien. Ber. 114 [2a], p. 1301, 1905. Ann. d. Phys. 36, p. 91, 1911. H. A. Lorentz, Phys. Zeitschr. 11, p. 1244, 1910. H. Poincare, Journ. de Phys. (5) 2, p. 5, p. 347, 1912.

AUTHOR'S BIBLIOGRAPHY

List of the papers published by the author on heat radiation and the hy pothesis of quanta, with references to the sections of this book where the same subject is treated.

Absorption und Emission elektrischer Wellen durch Resonanz. Sitzungs- ber. d. k. preuss. Akad. d. Wissensch. vom 21. Marz 1895, p. 289-301. WIED. Ann. 57, p. 1-14, 1896.

Ueber elektrische Schwingungen, welche durch Resonanz erregt und durch Strahlung gedampft werden. Sitzungsber. d. k. preuss. Akad. d. Wissensch. vom 20. Februar 1896, p. 151-170. WIED. Ann. 60, p. 577- 599, 1897.

Ueber irreversible Strahlungsvorgange. (Erste Mitteilung.) Sitzungs ber. d. k. preuss. Akad. d. Wissensch. vom 4. Februar 1897, p. 57-68.

Ueber irreversible Strahlungsvorgange. (Zweite Mitteilung.) Sitzungs ber. d. k. preuss. Akad. d. Wissensch. vom 8. Juli 1897, p. 715-717.

Ueber irreversible Strahlungsvorgange. (Dritte Mittelung.) Sitzungs ber. d. k. preuss. Akad. d. Wissensch. vom 16. Dezember 1897, p. 1122- 1145.

Ueber irreversible Strahlungsvorgange. (Vierte Mitteilung.) Sitzungs ber. d. k. preuss. Akad. d. Wissensch. vom 7. Juli 1898, p. 449-476.

Ueber irreversible Strahlungsvorgange. (Fiinfte Mitteilung.) Sitzungs ber. d. k. preuss. Akad. d. Wissensch. vom 18. Mai 1899, p. 440-480. (§§ 144 bis 190. § 164.)

Ueber irreversible Strahlungsvorgange. Ann. d. Phys. 1, p. 69-122, 1900. (§§ 144-190. § 164.)

Entropie und Temperatur strahlender Warme. Ann. d. Phys. 1, p. 719 bis 737, 1900. (§ 101. § 166.)

Ueber eine Verbesserung der WiENschen Spektralgleichung. Verhand- lungen der Deutschen Physikalischen Gesellschaft 2, p. 202-204, 1900. (§ 156.)

Ein vermeintlicher Widerspruch des magneto-optischen FARADAY- Effektes mit der Thermodynamik. Verhandlungen der Deutschen Phys ikalischen Gesellschaft 2, p. 206-210, 1900.

Kritikzweier Satze des Herrn W. WIEN. Ann. d. Phys. 3, p. 764-766, 1900.

Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verhandlungen der Deutschen Physikalischen Gesellschaft 2, p. 237-245, 1900. (§§141-143. §156f. §163.)

Ueber das Gesetz der Energieverteilung im Normalspektrum. Ann. d. Phys. 4, p. 553-563, 1901. (§§ 141-143. §§ 156-162.)

Ueber die Elementarquanta der Materie und der Elektrizitat. Ann. d Phys. 4, p. 564-566, 1901. (§ 163.)

Ueber irreversible Strahlungsvorgange (Nachtrag). Sitzungsber. d. k. preuss. Akad. d. Wissensch. vom 9. Mai 1901, p. 544-555. Ann. d. Phys. 6, p. 818-831, 1901. (§§ 185-189.)

Vereinfachte Ableitung der Schwingungsgesetze eines linearen Reson ators im stationar durchstrahlten Felde. Physikalische Zeitschrift 2, p. 530 bis p. 534, 1901.

Ueber die Natur des weissen Lichtes. Ann. d. Phys. 7, p. 390-400, 1902. (§§ 107-112. §§ 170-174.)

Ueber die von einem elliptisch schwingenden Ion emittierte und absorb-

216

AUTHOR'S BIBLIOGRAPHY 217

ierte Energie. Archives Ne*erlan daises, Jubelband fiir H. A. LOREXTZ, 1900, p. 164-174. Ann. d. Phys. 9, p. 619-628, 1902.

Ueber die Verteilung der Energie zwischen Aether und Materie. Archives Neerlandaises, Jubelband fur J. BOSSCHA, 1901, p. 55-66. Ann. d. Phys. 9, p. 629-641, 1902. (§§ 121-132.)

Bemerkimg liber die Konstante des WiENschen Verschiebimgsgesetzes. Verhandlungen der Deutschen Physikalischen Gesellschaft 8, p. 695-696, 1906. (§ 161.)

Zur Theorie der Warmestrahlung. Ann. d. Phys. 31, p. 758-768, 1910. Eine neue Strahlungshypothese. Verhandlungen der Deutschen Phys ikalischen Gesellschaft 13, p. 138-148, 1911. (§ 147.)

Zur Hypothese der Quantenemission. Sitzungsber. d. k. preuss. Akad. d. Wissensch. vom 13. Juli 1911, p. 723-731. (§§ 150-152.)

Ueber neuere thermodynamische Theorien (NERNSTsches Warmetheorem und Quantenhypothese-) . Ber. d: Deutschen Chemischen Gesellschaft 45, p. 5-23, 1912. Physikalische Zeitschrift 13, p. 165-175, 1912. Akadem- ische Verlagsgesellschaft m. b. H., Leipzig 1912. (§§ 120-125.)

Ueber die Begriindung des Gesetzes der schwarzen Strahlung. Ann. d. Phys. 37, p. 642-656, 1912. (§§ 145-156.)

APPENDIX I

On Deductions from Stirling's Formula. The formula is

nf (a) lim — ' -- =1,

n= «>nne-nV2Trn

or, to an approximation quite sufficient for all practical purposes, provided that n is larger than 7

(b)

For a proof of this relation and a discussion of its limits of accuracy a treatise on probability must be consulted. On substitution in (170) this gives

(N\»

W

On account of (165) this reduces at once to

NN

' V27TA/-2 . . . Passing now to the logarithmic expression we get S = k log W = k[N logN-Ni logNi-Nz logAT2-

+log V27rJV-lo or,

(AT2log]V2-log Now, for a large value of Ni} the term Nt log Ni is very much larger than log ^2-jrNi, as is seen by writing the latter in the form J log 2ir +} log JVf. Hence the last expression will, with a fair approximation, reduce to

S = k\og W = k[N\og N-NtlogNi-Nzlog N2- ..... ].

218

APPENDIX I 219

Introducing now the values of the densities of distribution w by means of the relation

Ni=*WiN we obtain

or, snce and hence and

N 1

log TV -log ATi=log -= log = -log

we obtain by substitution, after one or two simple transformations S = k log W= -kN2 wi log wi,

a relation which is identical with (173).

The statements of Sec. 143 may be proven in a similar manner. From (232) we get at once

S = k log Wm = k log ^T)/ -p

Now log (N - 1) / = log TV/ - log N,

and, for large values of N, log N is negligible compared with log N! Applying the same reasoning to the numerator we may without appreciable error write

Substituting now for (N+P)!, N!, and P! their values from (b) and omitting, as was previously shown to be approximately correct, the terms arising from the v2ir(N-}-P) etc., we get, since the terms containing e cancel out

S = k[(N+P) log (N+P)-N log N-P log P]

N --P

= k[(N+P) log ^ +P log N-P log P]

\ P

v+l)-jVlog

This is the relation of Sec. 143.

APPENDIX II

REFERENCES

Among general papers treating of the application of the theory of quanta to different parts of physics are :

1. A. Sommerfeld, Das Planck'sche Wirkungsquantum und seine allgemeine Bedeutung fiir die Molekularphysik, Phys. Zeitschr., 12, p. 1057. Report to the Versammlung Deutscher Naturforscher und Aerzte. Deals especially with applications to the theory of specific heats and to the photoelectric effect. Numerous references are quoted.

2. Meeting of the British Association, Sept., 1913. See Nature, 92, p. 305, Nov. 6, 1913, and Phys. Zeitschr., 14, p. 1297. Among the principal speakers were J. H. Jeans and H. A. Lorentz.

(Also American Phys. Soc., Chicago Meeting, 1913. l)

3. R. A. Millikan, Atomic Theories of Radiation, Science, 37, p. 119, Jan. 24, 1913. A non-mathematical discussion.

4. W. Wien, Neuere Probleme der Theoretischen Physik,

5. (Wien's  Columbia  Lectures,  in  German.)     This  is  perhaps 
   

the most complete review of the entire theory of quanta.

H. A. Lorentz, Alte und Neue Probleme der Physik, Phys. Zeitschr., 11, p. 1234. Address to the Versammlung Deutscher Naturforscher und Aerzte, Konigsberg, 1910, contains also some discussion of the theory of quanta.

Among the papers on radiation are :

E. Bauer, Sur la theorie du rayonnement, Comptes Rendus, 153, p. 1466. Adheres to the quantum theory in the original form, namely, that emission and absorption both take place in a discontinuous manner.

E. Buckingham, Calculation of 02 in Planck's equation, Bull. Bur. Stand. 7, p. 393.

E. Buckingham, On Wien's Displacement Law, Bull. Bur. Stand. 8, p. 543. Contains a very simple and clear proof of the displacement law.

'.Not yet published (Jan. 26, 1914. Tr.)

220

APPENDIX II 221

P. Ehrenfest, Strahlungshypothesen, Ann. d. Phys., 36, p. 91.

A. Joffe, Theorie der Strahlung, Ann. d. Phys., 36, p. 534.

Discussions of the method of derivation of the radiation formula are given in many papers on the subject. In addition to those quoted elsewhere may! ;be maeatioiied :

C. Benedicks, Ueber die Herleitung von Planck's Energiever- teilungsgesetz, Ann. d. Phys., 42, p. 133. Derives Planck 's law without the help of the quantum theory. The law of equiparti- tion of energy is avoided by the assumption that solids are not always monatomic, but that, with decreasing temperature, the atoms form atomic complexes, thus changing the number of degrees of freedom. The equipartition principle applies only to the free atoms.

P. Debye, Planck7 s Strahlungsformel, Ann. d. Phys., 33, p. 1427. This method is fully discussed by Wien (see 4, above). It somewhat resembles Jeans7 method (Sec. 169) since it avoids all reference to resonators of any particular kind and merely establishes the most probable energy distribution. It differs, however, from Jeans' method by the assumption of discrete energy quanta hv. The physical nature of these units is not discussed at all and it is also left undecided whether it is a property of matter or of the ether or perhaps a property of the energy exchange between matter and the ether that causes their existence. (Compare also some remarks of Lorentz in 2.)

P. Frank, Zur Ableitung der Planckschen Strahlungsformel, Phys. Zeitschr., 13, p. 506.

L. Natanson, Statistische Theorie der Strahlung, Phys. Zeitschr., 12, p. 659.

W. Nernst, Zur Theorie der Specifischen Warme und iiber die Anwendung, der Lehre von den Energiequanten auf Physikalisch- chemische Fragen uberhaupt, Zeitschr. f. Elektochemie, 17, p. 265.

The experimental facts on which the recent theories of specific heat (quantum theories) rely, were discovered by W. Nernst and his fellow workers. The results are published in a large number of papers that have appeared in different periodicals. See, e.g., W. Nernst, Der Energieinhalt fester Substanzen, Ann. d. Phys., 36, p. 395, where also numerous other papers are quoted. (See also references given in 1.) These experimental facts give very strong support to the heat theorem of Nernst (Sec. 120),

222 APPENDIX II

according to which the entropy approaches a definite limit (perhaps the value zero, see Planck's Thermodynamics, 3. ed., sec. 282, et seq.) at the absolute zero of temperature, and which is consistent with the quantum theory!;^ This work is in close connection with the recent latfeeiiilpt&Tto-'idevelop an equation of state applicable to the solid state r of matter. In addition to the papers by Nernst and his school there may be mentioned :

K. Eisenmann, Canonische Zustandsgleichung einatomiger fester Korper und die Quantentheorie, Verhandlungen der Deutschen Physikalischen Gesellschaft, 14, p. 769.

W. H. Keesom, Entropy and the Equation of State, Konink. Akad. Wetensch. Amsterdam Proc., 15, p. 240.

L. Natanson, Energy Content of Bodies, Acad. Science Cra- covie Bull. Ser. A, p. 95. In Einstein's theory of specific heats (Sec. 140) the atoms of actual bodies in nature are apparently identified with the ideal resonators of Planck. In this paper it is pointed out that this is implying too special features for the atoms of real bodies, and also, that such far-reaching specializa tions do not seem necessary for deriving the laws of specific heat from the quantum theory.

L. S. Ornstein, Statistical Theory of the Solid State, Konink. Akad. Wetensch. Amsterdam Proc., 14, p. 983.

S. Ratnowsky, Die Zustandsgleichung einatomiger fester Korper und die Quantentheorie, Ann. d. Phys., 38, p. 637.

Among papers on the law of equipartition of energy (Sec. 169) are:

J. H. Jeans, Planck's Radiation Theory and Non-Newtonian Mechanics, Phil. Mag., 20, p. 943.

S. B. McLaren, Partition of Energy between Matter and Radiation, Phil. Mag., 21, p. 15.

S. B. McLaren, Complete Radiation, Phil. Mag. 23, p. 513. This paper and the one of Jeans deal with the fact that from Newtonian Mechanics (Hamilton's Principle) the equipartition principle necessarily follows, and that hence either Planck's law or the fundamental principles of mechanics need a modification.

For the law of equipartition compare also the discussion at the meeting of the British Association (see 2).

In many of the papers cited so far deductions from the quan-

APPENDIX II 223

turn theory are compared with experimental facts. This is also done by :

F. Haber, Absorptionsspectra fester Korper und die Quanten- theorie, Verhandlungen der Deutschen Physikalischen Gesell- schaft, 13, p. 1117.

J. Franck und G. Hertz, Quantumhypothese und lonisation, Ibid., 13, p. 967.

Attempts of giving a concrete physical idea of Planck's con stant h are made by :

A. Schidlof, Zur Aufklarung der universellen electrodyna- mischen Bedeutung der Planckschen Strahlungsconstanten h, Ann. d. Phys., 35, p. 96.

D. A. Goldhammer, Ueber die Lichtquantenhypothese, Phys. Zeitschr., 13, p. 535.

J. J. Thomson, On the Structure of the Atom, Phil. Mag., 26, p. 792.

N. Bohr, On the Constitution of the Atom, Phil. Mag., 26, p. 1.

S. B. McLaren, The Magneton and Planck's Universal Con stant, Phil. Mag., 26, p. 800.

The line of reasoning may be briefly stated thus : Find some quantity of the same dimension as h, and then construct a model of an atom where this property plays an important part and can be made, by a simple hypothesis, to vary by finite amounts in stead of continuously. The simplest of these is Bohr's, where h is interpreted as angular momentum.

The logical reason for the quantum theory is found in the fact that the Rayleigh- Jeans radiation formula does not agree with experiment. Formerly Jeans attempted to reconcile theory and experiment by the assumption that the equilibrium of radiation and a black body observed and agreeing with Planck' 's law rather than his own, was only apparent, and that the true state of equilibrium which really corresponds to his law and the equipartition of energy among all variables, is so slowly reached that it is never actually observed. This standpoint, which was strongly objected to by authorities on the experi mental side of the question (see, e.g., E. Pringsheim in 2), he has recently abandoned. H. Poincare, in a profound mathe matical investigation (H. Poincare, Sur. la Theorie des Quanta,

224 APPENDIX II

Journal de Physique (5), 2, p. 1, 1912) reached the conclusion that whatever the law of radiation may be, it must always, if the total radiation is assumed as finite, lead to a function pre senting similar discontinuites as the one obtained from the hypothesis of quanta.

While most authorities have accepted the quantum theory for good (see J. H. Jeans and H. A. Lorentz in 2), a few still enter tain doubts as to the general validity of Poincare's conclusion (see above C. Benedicks and R. A Millikan 3). Others still reject the quantum theory on account of the fact that the ex perimental evidence in favor of Planck's law is not absolutely conclusive (see R. A. Millikan 3); among these is A. E. H. Love (2), who suggests that Korn's (A. Korn, Neue Mechanische Vorstellungen uber die Schwarze Strahlung und eine sich aus denselben ergebende Modification des Planckschen Verteilungs- gesetzes, Phys. Zeitschr., 14, p. 632) radiation formula fits the facts as well as that of Planck.

H. A. Callendar, Note on Kadiation and Specific Heat, Phil. Mag., 26, p. 787, has also suggested a radiation formula that fits the data well. Both Korn's and Callendar's formulae conform to Wien's displacement law and degenerate for large values of \T into the Rayleigh-Jeans, and for small values of \T into Wien's radiation law. Whether Planck's law or one of these is the correct law, and whether, if either of the others should prove to be right, it would eliminate the necessity of the adop tion of the quantum theory, are questions as yet undecided. Both Korn and Callendar have promised in their papers to follow them by further ones.

ERRATA

Page 77. The last sentence of Sec. 77 should be replaced by: The corresponding additional terms may, however, be omitted here without appreciable error, since the correction caused by them would consist merely of the addition to the energy change here calculated of a comparatively infinitesimal energy change of the same kind with an external work that is infinitesimal of the second order.

Page 83. Insert at the end of Sec. 84 a:

These laws hold for any original distribution of energy what ever; hence, e. g., an originally monochromatic radiation remains monochromatic during the process described, its color changing in the way stated.

225

lETURN Astronomy/Mathematics/Statistics/Computer Science Library Qm+ 100 Evans Ha 1 1 642-3381

OANPERlOm

2

3

5

1 MOWTM

6

-w — <- m v y

ALL BOOKS MAY BE RECALLED AFTER 7 DAYS

DUE AS STAMPED BELOW

M^¥ j ff i^s

0

ORM NO. DD 3, 13m/6/76

UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720

QC33I