The Theory of Heat Radiation (1914) — part 11 of 12

where Cv (positive) and 6V denote certain functions of the posi" tive variable of integration v. The values of these functions are not wholly determined by the behavior of E2 in the time interval mentioned, but depend also on the manner in which Ez varies as a function of the time beyond both ends of that interval. Hence the quantities Cv and Qv possess separately no definite physical significance, and it would be quite incorrect to think of the vibration Ez as, say, a continuous spectrum of periodic vibrations with the constant amplitudes Cv. This may, by the way, be seen at once from the fact that the character of the vibra tion Ez may vary with the time in any way whatever. How the spectral resolution of the vibration Ez is to be performed and to what results it leads will be shown below (Sec. 174).

172. We shall, as heretofore (158), define J, the " intensity of the exciting vibration/'1 as a function of the time to be the mean value of Ez2 in the time interval from t to t+r, where T is taken as large compared with the time 1/v, which is the duration of one of the periodic partial vibrations contained in the radiation, but as small as possible compared with the time T. In this statement there is a certain indefiniteness, from which results the fact that J will, in general, depend not only on t but also on T. If this is the case one cannot speak of the intensity of the exciting vibra tion at all. For it is an essential feature of the conception of the intensity of a vibration that its value should change but unap- preciably within the time required for a single vibration. (Com pare above, Sec. 3.) Hence we shall consider in future only those processes for which, under the conditions mentioned, there exists a mean value of Ez2 depending only on t. We are then obliged to assume that the quantities Cv in (311) are negligible for all

values of v which are of the same order of magnitude as - or smaller, i.e.,

vr is large. (312)

1 Not to be confused with the "field intensity" (field-strength) Ez of the exciting vibra tion.

192 IRREVERSIBLE RADIATION PROCESSES

In order to calculate J we now form from (311) the value of E22 and determine the mean value E22 of this quantity by inte grating with respect to t from t to t+T, then dividing by r and passing to the limit by decreasing r sufficiently. Thus we get

00 00

E,2= ( Cdvf dv Cv> Cv cos (27T/Z-0/) cos

-if"

Jo Jo

If we now exchange the values of v and /, the function under the sign of integration does not change; hence we assume

v'>v and write:

E*2 = 2 J J dv' dv C/ Cv cos (2irvft-e^ cos (2irvt-0,), or

dv' dv CV Cjcos

= f C

And hence

,, ,

[ TT(V'-V)T

sin TT (v'+ v) T -COS [TT(/+ v) (2^+r) - 0/- 0J \

If we now let r become smaller and smaller, since vr remains large, the denominator (V'+V)T of the second fraction remains large under all circumstances, while that of the first fraction (/— V)T may decrease with decreasing value of r to less than any finite value. Hence for sufficiently small values of v' —v the in tegral reduces to

dv' dv Cv> Cv cos [27r(/-^)^-0/+0J

which is in fact independent of r. The remaining terms of the double integral, which correspond to larger values of /— v, i.e., to more rapid changes with the time, depend in general on T and

FIELDS OF RADIATION IN GENERAL 193

therefore must vanish, if the intensity / is not to depend on r. Hence in our case on introducing as a second variable of integra tion instead of v

we have

J= \ \ d» dv C,+MC, cos (2-jrfjit- 0,+lt+0r) (313)

= \ \

= I

where A(Ji= I dv€v+tl.Cv cos (0,+M-0,,) (314)

or

J= dA cos 2irjLt--BJ sin

By this expression the intensity J of the exciting vibration, if it exists at all, is expressed by a function of the time in the form of a Fourier's integral.

173. The conception of the intensity of vibration J necessarily contains the assumption that this quantity varies much more slowly with the time t than the vibration Ez itself. The same follows from the calculation of J in the preceding paragraph. For there, according to (312), vr and v'r are large, but (/ — v}r is small for all pairs of values Cv and C/ that come into considera tion; hence, a fortiori,

^-^ = - is small, (315)

V V

and accordingly the Fourier's integrals E2 in (311) and J in (314) vary with the time in entirely different ways. Hence in the following we shall have to distinguish, as regards dependence on time, two kinds of quantities, which vary in different ways: Rapidly varying quantities, as E2, and slowly varying quantities as J and I the spectral intensity of the exciting vibration, whose value we shall calculate in the next paragraph. Nevertheless this difference in the variability with respect to time of the quanti-

13

194 IRREVERSIBLE RADIATION PROCESSES

ties named is only relative, since the absolute value of the differ ential coefficient of J with respect to time depends on the value of the unit of time and may, by a suitable choice of this unit, be made as large as we please. It is, therefore, not proper to speak of J(t) simply as a slowly varying function of t. If, in the following, we nevertheless employ this mode of expression for the sake of brevity, it will always be in the relative sense, namely, with respect to the different behavior of the function Eg(t).

On the other hand, as regards the dependence of the phase constant 6V on its index v it necessarily possesses the property of rapid variability in the absolute sense. For, although ^ is small compared with v, nevertheless the difference 0,,+/i — 6V is in general not small, for if it were, the quantities A^ and 5M in (314) would have too special values and hence it follows that (&Qv/'bv)'v must be large. This is not essentially modified by changing the unit of time or by shifting the origin of time.

Hence the rapid variability of the quantities 6V and also Cv with v is, in the absolute sense, a necessary condition for the existence of a definite intensity of vibration /, or, in other words, for the possibility of dividing the quantities depending on the time into those which vary rapidly and those which vary slowly — • a distinction which is also made in other physical theories and upon which all the following investigations are based.

174. The distinction between rapidly variable and slowly variable quantities introduced in the preceding section has, at the present stage, an important physical aspect, because in the following we shall assume that only slow variability with time is capable of direct measurement. On this assumption we approach conditions as they actually exist in optics and heat radiation. Our problem will then be to establish relations be tween slowly variable quantities exclusively; for these only can be compared with the results of experience. Hence we shall now determine the most important one of the slowly variable quanti ties to be considered here, namely, the "spectral intensity" I of the exciting vibration. This is effected as in (158) by means of the equation

\dv.

FIELDS OF RADIATION IN GENERAL By comparison with 313 we obtain:

cos 2i

195

where

= I 4t(A

sn

(316)

, cos

sn

By this expression the spectral intensity, I , of the exciting vibra tion at a point in the spectrum is expressed as a slowly variable function of the time t in the form of a Fourier's integral. The dashes over the expressions on the right side denote the mean values extended over a narrow spectral range for a given value of /*. If such mean values do not exist, there is no definite spec tral intensity.

CHAPTER II ONE OSCILLATOR IN THE FIELD OF RADIATION

175. If in any field of radiation whatever we have an ideal oscillator of the kind assumed above (Sec. 135), there will take place between it and the radiation falling on it certain mutual actions, for which we shall again assume the validity of the elementary dynamical law introduced in the preceding section. The question is then, how the processes of emission and absorp tion will take place in the case now under consideration.

In the first place, as regards the emission of radiant energy by the oscillator, this takes place, as before, according to the hypothe sis of emission of quanta (Sec. 147), where the probability quantity 77 again depends on the corresponding spectral intensity I through the relation (265).

On the other hand, the absorption is calculated, exactly as above, from (234), where the vibrations of the oscillator also take place according to the equation (233). In this way, by calculations analogous to those performed in the second chapter of the preceding part, with the difference only that instead of the Fourier's series (235) the Fourier's integral (311) is used, we obtain for the energy absorbed by the oscillator in the time r the expression

• I

-- C?M(,AM cos Zirpt+B^ sin

where the constants AM and BM denote the mean values expressed in (316), taken for the spectral region in the neighborhood of the natural frequency v0 of the oscillator. Hence the law of absorption will again be given by equation (249), which now holds also for an intensity of vibration I varying with the time. 176. There now remains the problem of deriving the expression for I, the spectral intensity of the vibration exciting the oscil lator, when the thermodynamic state of the field of radiation at

196

ONE OSCILLATOR IN THE FIELD OF RADIATION 197

the oscillator is given in accordance with the statements made in Sec. 17.

Let us first calculate ,the total intensity J = Ez2 of the vibration exciting an oscillator, from the intensities of the heat rays strik ing the oscillator from all directions.

For this purpose we must also allow for the polarization of the monochromatic rays which strike the oscillator. Let us begin by considering a pencil which strikes the oscillator within a con ical element whose vertex lies in the oscillator and whose solid angle, d!2, is given by (5), where the angles 6 and <£, polar coordi nates, designate the direction of the propagation of the rays. The whole pencil consists of a set of monochromatic pencils, one of which may have the principal values of intensity K and K' (Sec. 17). If we now denote the angle which the plane of vibration belonging to the principal intensity K makes with the plane through the direction of the ray and the 2-axis (the axis of the oscillator) by \p, no matter in which quadrant it lies, then, according to (8), the specific intensity of the monochromatic pencil may be resolved into the two plane polarized components at right angles with each other,

K cos2 ^ + K' sin K sin2 ^ + K' cos

the first of which vibrates in a plane passing through the 2-axis and the second in a plane perpendicular thereto.

The latter component does not contribute anything to the value of E22, since its electric field-strength is perpendicular to the axis of the oscillator. Hence there remains only the first

7T

component whose electric field-strength makes the angle — — 0

2

with the 2-axis. Now according to Poynting's law the intensity of

^i

a plane polarized ray in a vacuum is equal to the product of —

4ir

and the mean square of the electric field-strength. Hence the mean square of the electric field-strength of the pencil here considered is

— (K cos2 ^+ K' sin2 ^) dv dQ,

198 IRREVERSIBLE RADIATION PROCESSES

and the mean square of its component in the direction of the 2-axis is

47T

(K cos2 if'+K' sin2 ^) sin20 dv dfl. (317)

c

By integration over all frequencies and all solid angles we then obtain the value required

•   47T     C  C 
  

EZ2 = — I sin2 Bdtt I dv(Kv cos2 ^+K/ sin2 t)=J. (318)

The space density u of the electromagnetic energy at a point of the field is

u = ~ (E72 + E/+ E?+ HU+ H7+ H7) ,

O7T

where Ex2, E^2, E22, H*2, Hy2, H,2 denote the squares of the field-strengths, regarded as " slowly variable" quantities, and are hence supplied with the dash to denote their mean value. Since for every separate ray the mean electric and magnetic energies are equal, we may always write

If, in particular, all rays are unpolarized and if the intensity of radiation is constant in all directions, KJ/=K/ and, since

--_327r_2 C — —

3c J

and, by substitution in (319),

STrf

u=— I K,dv, c J

which agrees with (22) and (24).

177. Let us perform the spectral resolution of the intensity / according to Sec. 174; namely,

J =

ONE OSCILLATOR IN THE FIELD OF' RADIATION 199

Then, by comparison with (318), we find for the intensity of a definite frequency v contained in the exciting vibration the value

sin2 0 dQ(K, cos2 ^+K/ sin2 $). (320)

For radiation which is unpolarized and uniform in all directions we obtain again, in agreement with (160),

327T2

178. With the value (320) obtained for I the total energy absorbed by the oscillator in an element of time dt from the radiation falling on it is found from (249) to be

V I sin2 0 dtt(K cos2 ;//+K' sin2 cLJ

Hence the oscillator absorbs in the time dt from the pencil striking it within the conical element d£l an amount of energy equal to

sin2 0(K cos2 i£+K' sin2 $)dQ. (321)

cL

CHAPTER III A SYSTEM OF OSCILLATORS

179. Let us suppose that a large number N of similar oscillators with parallel axes, acting quite independently of one another, are distributed irregularly in a volume-element of the field of radia tion, the dimensions of which are so small that within it the inten sities of radiation K do not vary appreciably. We shall investi gate the mutual action between the oscillators and the radiation which is propagated freely in space.

As before, the state of the field of radiation may be given by the magnitude and the azimuth of vibration \f/ of the principal intensities Ky and K/ of the pencils which strike the system of oscillators, where Kv and K/ depend in an arbitrary way on the direction angles 6 and <£. On the other hand, let the state of the system of oscillators be given by the densities of distribution Wit W2, Wzj (166), with which the oscillators are dis tributed among the different region elements, Wi, w2) ws, . . . . being any proper fractions whose sum is 1. Herein, as always, the nth region element is supposed to contain the oscillators with energies between (n — )hv and nhv.

The energy absorbed by the system in the time dt within the conical element dtt is, according to (321),

sin2 0(K cos2 if>+ K' sin2 ^)dQ. (322)

cL

Let us now calculate also the energy emitted within the same conical element.

180. The total energy emitted in the time element dt by all N oscillators is found from the consideration that a single oscillator, according to (249), takes up an energy element hv during the time

(323) 200

A SYSTEM OF OSCILLATORS 201

and hence has a chance to emit once, the probability being rj. We shall assume that the intensity I of the exciting vibration does not change appreciably in the time r. Of the Nwn oscil lators which at the time t are in the nth region element a number Nwnrj will emit during the time r, the energy emitted by each being nhv. From (323) we see that the energy emitted by all oscillators during the time element dt is

dt Nrj\dt

Nwn 77 nh v— = 2nwn)

r 4L

or, according to (265),

(324)

From this the energy emitted within the conical element d$l may be calculated by considering that, in the state of thermo- dynamic equilibrium, the energy emitted in every conical element is equal to the energy absorbed and that, in the general case, the energy emitted in a certain direction is independent of the energy simultaneously absorbed. For the stationary state we have from (160) and (265)

*" 3C 1-77

= 32^ 77 (325)

and further from (271) and (265)

= r?(l-77)n-1, (326)

pi \i-|-/>i/

and hence

2nwn = ii2n(l-ii)*-l = -' (327)

»7

Thus the energy emitted (324) becomes

(328)

This is, in fact, equal to the total energy absorbed, as may be found by integrating the expression (322) over all conical ele ments dti and taking account of (325).

202 IRREVERSIBLE RADIATION PROCESSES

Within the conical element d$l the energy emitted or absorbed will then be

irNdt

cL

or, from (325), (327) and (268),

w '

sm

sin2 e d® dt>

and this is the general expression for the energy emitted by the system of oscillators in the time element dt within the conical element d!2, as is seen by comparison with (324).

181. Let us now, as a preparation for the following deductions, consider more closely the properties of the different pencils passing the system of oscillators. From all directions rays strike the volume-element that contains the oscillators; if we again consider those which come toward it in the direction (6, 4>) within the conical element dtt, the vertex of which lies in the volume-element, we may in. the first place think of them as being resolved into their monochromatic constituents, and then we need consider further only that one of these constituents which corresponds to the frequency v of the oscillators; for all other rays simply pass the oscillators without influencing them or being influenced by them. The specific intensity of a monochromatic ray of frequency v is

K+K'

where K and K' represent the principal intensities which we assume as non-coherent. This ray is now resolved into two com ponents according to the directions of its principal planes of vibration (Sec. 176). The first component,

passes by the oscillators and emerges on the other side with no change whatever. Hence it gives a plane polarized ray, which starts from the system of oscillators in the direction (0,0) within the solid angle d$l and whose vibrations are perpendicular to the axis of the oscillators and whose intensity is

K sin2 t+ K' cos2 $ = K". (330)

A SYSTEM OF OSCILLATORS 203

The second component,

K cos2 ^+K' sin2^,

polarized at right angles to the first consists again, according to Sec. 176, of two parts

(K cos2 i£+K' sin2 $) cos2 6 (331)

and

(K cos2 I//+K' sin2 $) sin2 0, (332)

of which the first passes by the system without any change, since its direction of vibration is at right angles to the axes of the oscil lators, while the second is weakened by absorption, say by the small fraction /3. Hence on emergence this component has only the intensity

(1 -0) (K cos2 ^+ K' sin2 t) sin2 0. (333)

It is, however, strengthened by the radiation emitted by the sys tem of oscillators (329), which has the value

0'(l-r?) 2nwn sin2 0, (334)

where /3' denotes a certain other constant, which depends only on the nature of the system and whose value is obtained at once from the condition that, in the state of thermodynamic equi librium, the loss is just compensated by the gain.

For this purpose we make use of the relations (325) and (327) corresponding to the stationary state, and thus find that the sum of the expressions (333) and (334) becomes just equal to (332); and thus for the constant j3f the following value is found :

,_ 3c _ hv*_ 13 l332TT2p13 c2 *

Then by addition of (331), (333) and (334) the total specific intensity of the radiation which emanates from the system of oscillators within the conical element dft, and whose plane of vibration is parallel to the axes of the oscillators, is found to be K'" = K cos

,_,,

jSsin2 0(Ke-(K cos2 ^+K' sin2 +)) k

where for the sake of brevity the term referring to the emis sion is written

hv3

-(1-77) 2nwn = K.. (336)

c2

204 IRREVERSIBLE RADIATION PROCESSES

Thus we finally have a ray starting from the system of oscil lators in the direction (0,0) within the conical element dtt and consisting of two components K" and K"' polarized perpendicu larly to each other, the first component vibrating at right angles to the axes of the oscillators.

In the state of thermodynamic equilibrium

a result which follows in several ways from the last equations.

182. The constant @ introduced above, a small positive num ber, is determined by the spacial and spectral limits of the radia tion influenced by the system of oscillators. If q denotes the cross-section at right angles to the direction of the ray, A v the spectral width of the pencil cut out of the total incident radiation by the system, the energy which is capable of absorption and which is brought to the system of oscillators within the conical element d& in the time dt is, according to (332) and (11),

gAKK cos2 ^+K' sin2 f) sin2 0 do dt. (337)

Hence the energy actually absorbed is the fraction /3 of this value. Comparing this with (322) we get

0 = — ?T- (338)

q-AvcL

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 mono chromatic rays which lie close to the natural frequency v of the oscillators, since the rest are not altered at all by the system.

The system is struck in the direction (0, 0) within the conical element dti 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

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 d!2. As a compensation there emerges from the system on the other side in the same direction (0, 0) a pencil polarized in some definite way, the intensity of which is given by the sum of the two com ponents K" and K'". By it an amount of energy

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 sub-

205

206 IRREVERSIBLE RADIATION PROCESSES

trading the first expression from the second and by integrating with respect to dtt. Thus we get

dt

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

cL

dQ sin20 (Ke-(K cos2 i//+K' sin2 ^)). (339)

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

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

00

(340)

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 r, given by (323), left this region; they have either passed into the (n+l)st region, or they have performed an emission at the boundary of the two regions. In compensa tion there have entered (1 — v))Nwn-i oscillators during the time r, that is, all oscillators which, at the time t, were in the (n — l)st region element, excepting such as have lost their energy by emission. Thus we obtain for the required change in the time dt

Ndwn = dtN((l-r1)wn-1-wn). (341)

T

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

= rjN.

CONSERVATION OF ENERGY AND INCREASE OF ENTROPY 207 Hence we have

Ndwi = — N(i) — wi).

T

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

,

1-77

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

(342)

--l)Wn-l-W»), (343)

and the energy change (340) of the system of oscillators becomes N\dt

4L

The sum 2 may be simplified by recalling that

co oo oo

2 nwn-i = 2 (n — l)wn-i+ S wn-\ i i i

1

. rl=S nwn--— i

Then we have

dE = — ^ (1-77 ?nwn) . (344)

4L i

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 rela tions (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,

208 IRREVERSIBLE RADIATION PROCESSES

retains during rectilinear propagation both its specific intensity and its entropy without change, even when reflected at the sur face, 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 v of the oscillators, since the rest are not altered at all by the system.

The system of oscillators is struck in the direction (0,0) within the conical element dtt 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 \l/ and-+^ respectively, which are

2

assumed to be non-coherent. According to (141) and Sec. 182 this pencil conveys the entropy

2A*>[L(K) + L(K')] dQ dt (345)

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 dtt. In compensation a pencil starts from the system on the other side in the same direction (6,<j>) within dQ, having the

components K" and K'" with the azimuth of vibration - 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

) + L(K0') (346)

where K0 and K0' 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:

K! = K sin2 ^+ K cos2 $ (1 -0 sin2 0) = K(l -/3 sin2 0 cos2^)

with the azimuth of vibration tg'

l-/3sin20

CONSERVATION OF ENERGY AND INCREASE OF ENTROPY 209 K2 = K' cos2 <H- K' sin2 ifr(l -0 sin2 0) = K'(l -0 sin2 5 sin2 ^)

cot2 \p

with the azimuth of vibration tg2 ^2 = 7- — r^; — ,

I—/? sin2 0

and,

K3 = /3 sin2 0 Ke

with the azimuth of vibration tg ^3 = 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

gA^LCKO + LCKoOldfi dt (347)

is added to the field of radiation in the time dt. All told, the en tropy 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 d!2, is

-/

)-L(K)-L(K')]. (348)

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= —kN2wn log wn. i

Hence the entropy change in the time dt is

oo

dS = — 'kN S log wn dw i

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

-(l-n)w«-i) log w«. (349)

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.

14

210 IRREVERSIBLE RADIATION PROCESSES

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

£;} =j{2K+/3sin2 d(Ke- K) ±/3 sin2 0(K«-K)},

and hence

K0=K+/3sin2 0(Ke-K), K0' = K.

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

gdQ{L(K,)-L(K)}

-/•

sin2 0(K.-K)

r

or, by (338) and (278),

dtt si

r

*s

c2K/'

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

' C I °°

; j dOsin^lKSC^-^^ J . i

TrkNdt chvL

where

r = l-i?. (350)

We now must prove that the expression

F

r i °°

I dQ sin2 ^ K2(wn-£wn-i

*s

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 6 and </>. The positive proper frac tion f is according to (350), (265), and (320) given by

— ^- = -3c2 - J K sin2 B dQ. (352)

The quantities Wi, w2, w3, are any positive proper

CONSERVATION OF ENERGY AND INCREASE OF ENTROPY 211 fractions whatever which, according to (167), satisfy the condition

2wn = l (353)

while, according to (342),

WO-^Y' (354)

Finally we have from (336)

/i.i/3/- oo

(355)

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 0 and 0,

which, with fixed values of f , Wi, w2, w$, and Ke, will

make F a minimum. The necessary condition for this is dF = 0, where according to (352)

•

SKsin2 0d 12 = 0.

This gives, by considering that the quantities w and f do not depend on 0 and 0, as a necessary condition for the minimum,

and it follows, therefore, that the quantity in brackets, and hence also K itself is independent of 0 and <£. That in this case F really has a minimum value is readily seen by forming the second varia tion

= I

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 0 and 0. Then it follows, by taking account of (319a), that

K = /n>3 T

212 IRREVERSIBLE RADIATION PROCESSES

and, by also substituting Ke from (355),

188. It now remains to prove that the sum

-^n-l) l<)gU>n-[(l-r)ra-l] l^logf, (356)

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 f , makes the sum $ a minimum. In this case 5 <£ = 0, or

CO

N dWn

lwn — C 5wn-i) log wn + ( wn — (Wn-i) - (o57)

wn

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

00

2 8wn = 0 and dw0 = 0. (358)

i

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

•i+-~J -- Id- fin- 1] log r) =0.

(359)

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

log wn-{ log wn+l+U^^::±- [(1-r) w-1] log r (360)

Wn

must be independent of n.

The solution of this functional equation is

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

$ = 0. (362)

CONSERVATION OF ENERGY AND INCREASE OF ENTROPY 213

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

//• ir in &

wn wn wn

i

where all terms containing the second variation d2wn have been omitted since their coefficients are, by (360), independent of n and since

i This gives, taking account of (361),

oo

25wn2

"

or

i That the sum which occurs here, namely,

' Vr £W WVF * V W^ P Wr2v*wV * v *" V ^ i /O £!O \

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

00

-«„ dWndwn+i , an+i

i

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

l — an an+i / l\2 4 — — «

or

•fcw-in^— v (364)

4(1 — an)

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

a, - , <*2 ^, «3 4_v' •

214 IRREVERSIBLE RADIATION PROCESSES

Continuing the procedure an remains always positive and less than a' = - ( 1 — V I — f ) . To prove the correctness of this state ment we show that, if it holds for «„, it holds also for an+i. We assume, therefore, that an is positive and <«'. Then from

(364) an+i is positive and <j^j ,y But— — - = «'.

Hence «n+i<a'. Now, since the assumption made does actu ally hold for n = l, it holds in general. The sum (363) is thus essentially positive and hence the value (362) of $ 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).