IV Conservation of Energy and Increase Of Entropy. Conclusion

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 $d Ω$ 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

q Δ ν (K + K') d Ω d t

to the system in the time

d t

; hence this energy is taken from the ﬁeld of radiation on the side of the rays arriving within

d Ω

. 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

q Δ ν (K'' + K''') d Ω d t,

is added to the ﬁeld of radiation. Hence, all told, the change in energy of the ﬁeld of radiation in the time

d t

is obtained by subtracting the ﬁrst expression from the second and by integrating with respect to

d Ω

. Thus we get

d t Δ ν \int (K'' + K''' - K - K') q d Ω,

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

\frac{π N d t}{c L} \int d Ω {sin}^{2} θ (K_{e} - (K {cos}^{2} ψ + K' {sin}^{2} ψ)) .

(339)

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

E = N h ν \sum_{1}^{\infty} (n - \frac{1}{2}) w_{n},

where the quantities

w_{n}

whose total sum is equal to

1

represent the densities of distribution characteristic of the state. Hence the energy change in the time

d t

d E = N h ν \sum_{1}^{\infty} (n - \frac{1}{2}) d w_{n} = N h ν \sum_{1}^{\infty} n d w_{n} .

(340)

To calculate $d w_{n}$ we consider the $n$ th 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 $(1 - η) N w_{n - 1}$ oscillators during the time $τ$ , that is, all oscillators which, at the time $t$ , were in the $(n - 1)$ st region element, excepting such as have lost their energy by emission. Thus we obtain for the required change in the time $d t$

N d w_{n} = \frac{d t}{τ} N ((1 - η) w_{n - 1} - w_{n}) .

(341)

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 (w_{1} + w_{2} + w_{3} + \dots) N = η N .

Hence we have

N d w_{1} = \frac{d t}{τ} N (η - w_{1}) .

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

w_{0} = \frac{η}{1 - η} .

(342)

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

d w_{n} = \frac{I d t}{4 h ν L} ((1 - η) w_{n - 1} - w_{n}),

(343)

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

d E = \frac{N I d t}{4 L} \sum_{1}^{\infty} n ((1 - η) w_{n - 1} - w_{n}) .

The sum

\sum

may be simpliﬁed by recalling that

\begin{array}{l} \sum_{1}^{\infty} n w_{n - 1} & = \sum_{1}^{\infty} (n - 1) w_{n - 1} + \sum_{1}^{\infty} w_{n - 1} \\ = \sum_{1}^{\infty} n w_{n} + w_{0} + 1 = \sum_{1}^{\infty} n w_{n} + \frac{1}{1 - η} . \end{array}

Then we have

d E = \frac{N I d t}{4 L} (1 - η \sum_{1}^{\infty} n w_{n}) .

(344)

This expression may be obtained more readily by considering that $d E$ 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 $d Ω$ 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 $\frac{π}{2} + ψ$ respectively, which are assumed to be non-coherent. According to (141) and Sec. 182 this pencil conveys the entropy

q Δ ν [L (K) + L (K')] d Ω d t

(345)

to the system of oscillators in the time $d t$ , 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 $d Ω$ . In compensation a pencil starts from the system on the other side in the same direction $(θ, φ)$ within $d Ω$ having the components $K''$ and $K'''$ with the azimuth of vibration $\frac{π}{2}$ 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 (K_{0}) + L (K')

(346)

where $K_{0}$ and $K'$ represent the principal intensities of the pencil.

For the calculation of $K_{0}$ 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_{1} = K {sin}^{2} ψ + K {cos}^{2} ψ (1 - β {sin}^{2} θ) = K (1 - β {sin}^{2} θ {cos}^{2} ψ)

with the azimuth of vibration

{tg}^{2} ψ_{1} = \frac{{tg}^{2} ψ}{1 - β {sin}^{2} θ}

K_{2} = K' {cos}^{2} ψ + K' {sin}^{2} ψ (1 - β {sin}^{2} θ) = K' (1 - β {sin}^{2} θ {sin}^{2} ψ)

with the azimuth of vibration

{tg}^{2} ψ_{2} = \frac{{cot}^{2} ψ}{1 - β {sin}^{2} θ}

, and,

K_{3} = β {sin}^{2} θ K_{e}

with the azimuth of vibration

tg ψ_{3} = 0

According to (147) these values give the principal intensities $K_{0}$ and $K'$ required and hence the entropy radiation (346). Thereby the amount of entropy

q Δ ν [L (K_{0}) + L (K')] d Ω d t

(347)

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

d t Δ ν \int q d Ω [L (K_{0}) + L (K') - 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 $d t$ . According to (173) the entropy at the time $t$ is

S = - k N \sum_{1}^{\infty} w_{n} log w_{n} .

Hence the entropy change in the time

d t

d S = - k N \sum_{1}^{\infty} log w_{n} d w_{n},

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

d S = \frac{N k I d t}{4 h ν L} \sum_{1}^{\infty} (w_{n} - (1 - η) w_{n - 1}) log w_{n} .

(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 $K' = K$ .

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

\begin{aligned} K_{0} \\ K' \end{aligned}\} = \frac{1}{2} {2 K + β {sin}^{2} θ (K_{e} - K) \pm β {sin}^{2} θ (K_{e} - K)},

and hence

K_{0} = K + β {sin}^{2} θ (K_{e} - K), K' = K .

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

d t Δ ν \int q d Ω {L (K_{o}) - L (K)} = d t Δ ν \int q d Ω β {sin}^{2} θ (K_{e} - K) \frac{d L (K)}{d K}

or, by (338) and (278),

= \frac{π k N d t}{h c ν L} \int d Ω {sin}^{2} θ (K_{e} - K) log (1 + \frac{h ν^{3}}{c^{2} K}) .

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 $d t$ is found to be equal to the expression

\frac{π k N d t}{c h ν L} \int d Ω {sin}^{2} θ {K \sum_{1}^{\infty} (w_{n} - ζ w_{n - 1}) log w_{n} + (K_{e} - K) log (1 + \frac{h ν^{3}}{c^{2} K})}

where

ζ = 1 - η .

(350)

We now must prove that the expression

\begin{matrix} F = \int d Ω {sin}^{2} θ {K \sum_{1}^{\infty} (w_{n} - ζ w_{n - 1}) log w_{n} \\ + (K_{e} - K) log (1 + \frac{h ν^{3}}{c^{2} K})} \end{matrix}

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

\frac{ζ}{1 - ζ} = \frac{3 c^{2}}{8 π h ν^{3}} \int K {sin}^{2} θ d Ω .

(352)

The quantities $w_{1}$ , $w_{2}$ , $w_{3}, \dots$ are any positive proper fractions whatever which, according to (167), satisfy the condition

\sum_{1}^{\infty} w_{n} = 1

(353)

while, according to (342),

w_{0} = \frac{1 - ζ}{ζ} .

(354)

Finally we have from (336)

K_{e} = \frac{h ν^{3} ζ}{c^{2}} \sum_{1}^{\infty} n w_{n} .

(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 $θ$ and $φ$ , which, with ﬁxed values of $ζ$ , $w_{1}$ , $w_{2}$ , $w_{3}, \dots$ and $K_{e}$ , will make $F$ a minimum. The necessary condition for this is $δ F = 0$ , where according to (352)

\int δ K {sin}^{2} θ d Ω = 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,

δ F = 0 = \int d Ω {sin}^{2} θ δ K \{- log (1 + \frac{h ν^{3}}{c^{2} K}) - \frac{K_{e} - K}{\frac{c^{2} K}{h ν^{3}} + 1} \cdot \frac{1}{K}\}

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

δ^{2} F = \int d Ω {sin}^{2} θ δ K δ \{- log (1 + \frac{h ν^{3}}{c^{2} K}) - \frac{K_{e} - K}{\frac{c^{2} K}{h ν^{3}} + 1} \cdot \frac{1}{K}\}

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 = \frac{h ν^{3}}{c^{2}} \frac{ζ}{1 - ζ}

and, by also substituting

K_{e}

from (355),

F = \frac{8 π h ν^{3}}{3 c^{2}} \frac{ζ}{1 - ζ} \sum_{1}^{\infty} (w_{n} - ζ w_{n - 1}) log w_{n} - [(1 - ζ) n - 1] w_{n} log ζ .

188. It now remains to prove that the sum

Φ = \sum_{1}^{\infty} (w_{n} - ζ w_{n - 1}) log w_{n} - [(1 - ζ) n - 1] w_{n} log ζ,

(356)

where the quantities $w_{n}$ 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 $δ Φ = 0$ , or

\begin{matrix} \sum_{1}^{\infty} (δ w_{n} - ζ δ w_{n - 1}) log w_{n} + (w_{n} - ζ w_{n - 1}) \frac{δ w_{n}}{w_{n}} \\ - [(1 - ζ) n - 1] δ w_{n} log ζ = 0, & (IV.1) \end{matrix}

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

\sum_{1}^{\infty} δ w_{n} = 0 and δ w_{0} = 0 .

(358)

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

\sum_{1}^{\infty} δ w_{n} {log w_{n} - ζ log w_{n + 1} + \frac{w_{n} - ζ w_{n - 1}}{w_{n}} - [(1 - ζ) n - 1] log ζ} = 0 .

(359)

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

log w_{n} - ζ log w_{n + 1} + \frac{w_{n} - ζ w_{n - 1}}{w_{n}} - [(1 - ζ) n - 1] log ζ

(360)

must be independent of $n$ .

The solution of this functional equation is

w_{n} = (1 - ζ) ζ^{n - 1}

(361)

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

Φ = 0 .

(362)

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

δ^{2} Φ = \sum_{1}^{\infty} (δ w_{n} - ζ δ w_{n - 1}) \frac{δ w_{n}}{w_{n}} - \frac{ζ δ w_{n - 1}}{w_{n}} δ w_{n} + \frac{ζ w_{n - 1}}{w_{n}^{2}} δ w_{n}^{2},

where all terms containing the second variation

δ^{2} w_{n}

have been omitted since their coeﬃcients are, by (360), independent of

n

and since

\sum_{1}^{\infty} δ^{2} w_{n} = 0 .

This gives, taking account of (361),

δ^{2} Φ = \sum_{1}^{\infty} \frac{2 δ w_{n}^{2}}{(1 - ζ) ζ^{n - 1}} - \frac{2 ζ δ w_{n - 1} δ w_{n}}{(1 - ζ) ζ^{n - 1}}

δ^{2} Φ = \frac{2 ζ}{1 - ζ} \sum_{1}^{\infty} \frac{δ w_{n}^{2}}{ζ^{n}} - \frac{δ w_{n - 1} δ w_{n}}{ζ^{n - 1}} .

That the sum which occurs here, namely,

\frac{δ w_{1}^{2}}{ζ} - \frac{δ w_{1} δ w_{2}}{ζ} + \frac{δ w_{2}^{2}}{ζ^{2}} - \frac{δ w_{2} δ w_{3}}{ζ^{2}} + \frac{δ w_{3}^{2}}{ζ^{3}} - \frac{δ w_{3} δ w_{4}}{ζ^{3}} + \dots

(363)

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

\sum_{1}^{\infty} \frac{1 - α_{n}}{ζ^{n}} δ w_{n}^{2} - \frac{δ w_{n} δ w_{n + 1}}{ζ^{n}} + \frac{α_{n + 1}}{ζ^{n + 1}} δ w_{n + 1}^{2},

which is identical with (363) provided

α_{1} = 0

. Now the

α

’s may be so determined that every term of the last sum is a perfect square, i.e., that

4 \cdot \frac{1 - α_{n}}{ζ^{n}} \cdot \frac{α_{n + 1}}{ζ^{n + 1}} = {(\frac{1}{ζ^{n}})}^{2}

α_{n + 1} = \frac{ζ}{4 (1 - α_{n})} .

(364)

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

α_{1} = 0, α_{2} = \frac{ζ}{4}, α_{3} = \frac{ζ}{4 - ζ}, \dots .

Continuing the procedure

α_{n}

remains always positive and less than

α' = \frac{1}{2} (1 - \sqrt{1 - ζ})

. To prove the correctness of this statement we show that, if it holds for

α_{n}

, it holds also for

α_{n + 1}

. We assume, therefore, that

α_{n}

is positive and

< α'

. Then from (364)

α_{n + 1}

is positive and

< \frac{ζ}{4 (1 - α')}

. But

\frac{ζ}{4 (1 - α')} = α'

. Hence

α_{n + 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.

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Chapter IVConservation of Energy and Increase Of Entropy. Conclusion

Chapter IV
Conservation of Energy and Increase Of Entropy. Conclusion