Chapter IV

Conservation of Energy and Increase Of Entropy. Conclusion

183. It is now easy to state the relation of the two principles of thermodynamics to the irreversible processes here considered. Let us consider first the conservation of energy. If there is no oscillator in the field, every one of the elementary pencils, infinite in number, retains, during its rectilinear propagation, both its specific intensity K and its energy without change, even though it be reflected at the surface, assumed as plane and reflecting, which bounds the field (Sec. 166). The system of oscillators, on the other hand, produces a change in the incident pencils and hence also a change in the energy of the radiation propagated in the field. To calculate this we need consider only those monochromatic rays which lie close to the natural frequency ν of the oscillators, since the rest are not altered at all by the system.

The system is struck in the direction (θ,φ) within the conical element dΩ which converges toward the system of oscillators by a pencil polarized in some arbitrary way, the intensity of which is given by the sum of the two principal intensities K and K’. This pencil, according to Sec. 182, conveys the energy

qΔν(K+K)dΩdt

to the system in the time dt; hence this energy is taken from the field of radiation on the side of the rays arriving within dΩ. As a compensation there emerges from the system on the other side in the same direction (θ,φ) a pencil polarized in some definite way, the intensity of which is given by the sum of the two components K and K. By it an amount of energy

qΔν(K+K)dΩdt,

is added to the field of radiation. Hence, all told, the change in energy of the field of radiation in the time dt is obtained by subtracting the first expression from the second and by integrating with respect to dΩ. Thus we get

dtΔν(K+KKK)qdΩ,

or by taking account of (330), (335), and (338)

(339)

πNdtcLdΩ sin2θ(Ke(K cos2ψ+K sin2ψ)).

184. Let us now calculate the change in energy of the system of oscillators which has taken place in the same time dt. According to (219), this energy at the time t is

E=Nhν1(n12)wn,

where the quantities wn whose total sum is equal to 1 represent the densities of distribution characteristic of the state. Hence the energy change in the time dt is

(340)

dE=Nhν1(n12)dwn=Nhν1ndwn.

To calculate dwn we consider the nth region element. All of the oscillators which lie in this region at the time t have, after the lapse of time τ, given by (323), left this region; they have either passed into the (n+1)st region, or they have performed an emission at the boundary of the two regions. In compensation there have entered (1η)Nwn1 oscillators during the time τ, that is, all oscillators which, at the time t, were in the (n1)st region element, excepting such as have lost their energy by emission. Thus we obtain for the required change in the time dt

(341)

Ndwn=dtτN((1η)wn1wn).

A separate discussion is required for the first region element n=1. For into this region there enter in the time τ all those oscillators which have performed an emission in this time. Their number is

η(w1+w2+w3+)N=ηN.

Hence we have

Ndw1=dtτN(ηw1).

We may include this equation in the general one (341) if we introduce as a new expression

(342)

w0=η1η.

Then (341) gives, substituting τ from (323),

(343)

dwn=Idt4hνL((1η)wn1wn),

and the energy change (340) of the system of oscillators becomes

dE=NIdt4L1n((1η)wn1wn).

The sum may be simplified by recalling that

1nwn1=1(n1)wn1+1wn1=1nwn+w0+1=1nwn+11η.

Then we have

(344)

dE=NIdt4L(1η1nwn).

This expression may be obtained more readily by considering that dE is the difference of the total energy absorbed and the total energy emitted. The former is found from (250), the latter from (324), by taking account of (265).

The principle of the conservation of energy demands that the sum of the energy change (339) of the field of radiation and the energy change (344) of the system of oscillators shall be zero, which, in fact, is quite generally the case, as is seen from the relations (320) and (336).

185. We now turn to the discussion of the second principle, the principle of the increase of entropy, and follow closely the above discussion regarding the energy. When there is no oscillator in the field, every one of the elementary pencils, infinite in number, retains during rectilinear propagation both its specific intensity and its entropy without change, even when reflected at the surface, assumed as plane and reflecting, which bounds the field. The system of oscillators, however, produces a change in the incident pencils and hence also a change in the entropy of the radiation propagated in the field. For the calculation of this change we need to investigate only those monochromatic rays which lie close to the natural frequency ν of the oscillators, since the rest are not altered at all by the system.

The system of oscillators is struck in the direction (θ,φ) within the conical element dΩ converging toward the system by a pencil polarized in some arbitrary way, the spectral intensity of which is given by the sum of the two principal intensities K and K with the azimuth of vibration ψ and π2+ψ respectively, which are assumed to be non-coherent. According to (141) and Sec. 182 this pencil conveys the entropy

(345)

qΔν[L(K)+L(K)]dΩdt

to the system of oscillators in the time dt, where the function L(K) is given by (278). Hence this amount of entropy is taken from the field of radiation on the side of the rays arriving within dΩ. In compensation a pencil starts from the system on the other side in the same direction (θ,φ) within dΩ having the components K and K with the azimuth of vibration π2 and 0 respectively, but its entropy radiation is not represented by L(K)+L(K), since K and K are not non-coherent, but by

(346)

L(K0)+L(K)

where K0 and K represent the principal intensities of the pencil.

For the calculation of K0 and K we make use of the fact that, according to (330) and (335), the radiation K and K, of which the component K vibrates in the azimuth 0, consists of the following three components, non-coherent with one another:

K1=K sin2ψ+K cos2ψ(1β sin2θ)=K(1β sin2θ cos2ψ)

with the azimuth of vibration tg2ψ1= tg2ψ1β sin2θ,

K2=K cos2ψ+K sin2ψ(1β sin2θ)=K(1β sin2θ sin2ψ)

with the azimuth of vibration tg2ψ2= cot2ψ1β sin2θ, and,

K3=β sin2θKe

with the azimuth of vibration tgψ3=0.

According to (147) these values give the principal intensities K0 and K required and hence the entropy radiation (346). Thereby the amount of entropy

(347)

qΔν[L(K0)+L(K)]dΩdt

is added to the field of radiation in the time dt. All told, the entropy change of the field of radiation in the time dt, as given by subtraction of the expression (345) from (347) and integration with respect to dΩ, is

(348)

dtΔνqdΩ[L(K0)+L(K)L(K)L(K)].

Let us now calculate the entropy change of the system of oscillators which has taken place in the same time dt. According to (173) the entropy at the time t is

S=kN1wn logwn.

Hence the entropy change in the time dt is

dS=kN1 logwndwn,

and, by taking account of (343), we have:

(349)

dS=NkIdt4hνL1(wn(1η)wn1) logwn.

186. The principle of increase of entropy requires that the sum of the entropy change (348) of the field of radiation and the entropy change (349) of the system of oscillators be always positive, or zero in the limiting case. That this condition is in fact satisfied we shall prove only for the special case when all rays falling on the oscillators are unpolarized, i.e., when K=K.

In this case we have from (147) and Sec. 185.

K0K=12{2K+β sin2θ(KeK)±β sin2θ(KeK)},

and hence

K0=K+β sin2θ(KeK),K=K.

The entropy change (348) of the field of radiation becomes

dtΔνqdΩ{L(Ko)L(K)}=dtΔνqdΩβ sin2θ(KeK)dL(K)dK

or, by (338) and (278),

=πkNdthcνLdΩ sin2θ(KeK) log1+hν3c2K.

On adding to this the entropy change (349) of the system of oscillators and taking account of (320), the total increase in entropy in the time dt is found to be equal to the expression

πkNdtchνLdΩ sin2θ{K1(wnζwn1) logwn+(KeK) log(1+hν3c2K)}

where

(350)

ζ=1η.

We now must prove that the expression

F=dΩ sin2θ{K1(wnζwn1) logwn+(KeK) log1+hν3c2K}

is always positive and for that purpose we set down once more the meaning of the quantities involved. K is an arbitrary positive function of the polar angles θ and φ. The positive proper fraction ζ is according to (350), (265), and (320) given by

(352)

ζ1ζ=3c28πhν3K sin2θdΩ.

The quantities w1w2, w3, are any positive proper fractions whatever which, according to (167), satisfy the condition

(353)

1wn=1

while, according to (342),

(354)

w0=1ζζ.

Finally we have from (336)

(355)

Ke=hν3ζc21nwn.

187. To give the proof required we shall show that the least value which the function F can assume is positive or zero. For this purpose we consider first that positive function, K, of θ and φ, which, with fixed values of ζ, w1w2, w3, and Ke, will make F a minimum. The necessary condition for this is δF=0, where according to (352)

δK sin2θdΩ=0.

This gives, by considering that the quantities w and ζ do not depend on θ and φ, as a necessary condition for the minimum,

δF=0=dΩ sin2θδK log1+hν3c2KKeKc2Khν3+11K

and it follows, therefore, that the quantity in brackets, and hence also K itself is independent of θ and φ. That in this case F really has a minimum value is readily seen by forming the second variation

δ2F=dΩ sin2θδKδ log1+hν3c2KKeKc2Khν3+11K

which may by direct computation be seen to be positive under all circumstances.

In order to form the minimum value of F we calculate the value of K, which, from (352), is independent of θ and φ. Then it follows, by taking account of (319a), that

K=hν3c2ζ1ζ

and, by also substituting Ke from (355),

F=8πhν33c2ζ1ζ1(wnζwn1) logwn[(1ζ)n1]wn logζ.

188. It now remains to prove that the sum

(356)

Φ=1(wnζwn1) logwn[(1ζ)n1]wn logζ,

where the quantities wn are subject only to the restrictions that (353) and (354) can never become negative. For this purpose we determine that system of values of the w’s which, with a fixed value of ζ, makes the sum Φ a minimum. In this case δΦ=0, or

(IV.1)

1(δwnζδwn1) logwn+(wnζwn1)δwnwn[(1ζ)n1]δwn logζ=0,

where, according to (353) and (354),

(358)

1δwn=0 andδw0=0.

If we suppose all the separate terms of the sum to be written out, the equation may be put into the following form:

(359)

1δwn{logwnζ logwn+1+wnζwn1wn[(1ζ)n1] logζ}=0.

From this, by taking account of (358), we get as the condition for a minimum, that

(360)

logwnζ logwn+1+wnζwn1wn[(1ζ)n1] logζ

must be independent of n.

The solution of this functional equation is

(361)

wn=(1ζ)ζn1

for it satisfies (360) as well as (353) and (354). With this value (356) becomes

(362)

Φ=0.

189. In order to show finally that the value (362) of Φ is really the minimum value, we form from (357) the second variation

δ2Φ=1(δwnζδwn1)δwnwnζδwn1wnδwn+ζwn1wn2δwn2,

where all terms containing the second variation δ2wn have been omitted since their coefficients are, by (360), independent of n and since

1δ2wn=0.

This gives, taking account of (361),

δ2Φ=12δwn2(1ζ)ζn12ζδwn1δwn(1ζ)ζn1

or

δ2Φ=2ζ1ζ1δwn2ζnδwn1δwnζn1.

That the sum which occurs here, namely,

(363)

δw12ζδw1δw2ζ+δw22ζ2δw2δw3ζ2+δw32ζ3δw3δw4ζ3+

is essentially positive may be seen by resolving it into a sum of squares. For this purpose we write it in the form

11αnζnδwn2δwnδwn+1ζn+αn+1ζn+1δwn+12,

which is identical with (363) provided α1=0. Now the α’s may be so determined that every term of the last sum is a perfect square, i.e., that

41αnζnαn+1ζn+1=1ζn2

or

(364)

αn+1=ζ4(1αn).

By means of this formula the α’s may be readily calculated. The first values are:

α1=0,α2=ζ4,α3=ζ4ζ,.

Continuing the procedure αn remains always positive and less than α=1211ζ. To prove the correctness of this statement we show that, if it holds for αn, it holds also for αn+1. We assume, therefore, that αn is positive and <α. Then from (364) αn+1 is positive and <ζ4(1α). But ζ4(1α)=α. Hence αn+1<α. Now, since the assumption made does actually hold for n=1, it holds in general. The sum  (363) is thus essentially positive and hence the value (362) of Φ really is a minimum, so that the increase of entropy is proven generally.

The limiting case (361), in which the increase of entropy vanishes, corresponds, of course, to the case of thermodynamic equilibrium between radiation and oscillators, as may also be seen directly by comparison of (361) with (271), (265), and (360).

190. Conclusion.—The theory of irreversible radiation processes 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 general, 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 distribution 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 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.68

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 emission 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.

Chapter IV

68 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. Poincaré, Journ. de Phys. (5) 2, p. 5, p. 347, 1912.