The Theory of Heat Radiation (1914) — part 7 of 12

for all values of v entering into consideration. From this we must conclude that all amplitudes Cn with a moderately large value for the ordinal number n do not appear at all in the Fourier's series, that is to say, they are negligibly small.

109. Though we have no detailed special information about the function Ez, nevertheless its relation to the radiation of heat affords some important information as to a few of its general properties. Firstly, for the space density of radiation in a vacuum we have, according to Maxwell's theory,

Now the radiation is uniform in all directions and in the stationary

STATIONARY FIELD OF RADIATION 105

state, hence the six mean values named are all equal to one another, and it follows that

u=~^\ (151)

4-7T

Let us substitute in this equation the value of Ez as given by (149). Squaring the latter and integrating term by term through a time interval, from 0 to t, assumed large in comparison with all

periods of vibration ~ but otherwise arbitrary, and then divid- v

ing by t, we obtain, since the radiation is perfectly stationary,

«= -

From this relation we may at once draw an important conclu sion as to the nature of Ez as a function of time. Namely, since the Fourier's series (149) consists, as we have seen, of a great many terms, the squares, Cn2, of the separate amplitudes of vibration the sum of which gives the space density of radiation, must have exceedingly small values. Moreover in the integral of the square of the Fourier's series the terms which depend on the time t and contain the products of any two different amplitudes all cancel; hence the amplitudes Cn and the phase-constants 0n must vary from one ordinal number to another in a quite irregular manner. We may express this fact by saying that the separate partial vibrations of the series are very small and in a " chaotic"1 state.

For the specific intensity of the radiation travelling in any direction whatever we obtain from (21)

110. Let us now perform the spectral resolution of the last two equations. To begin with we have from (22) :

(154)

On the right side of the equation the sum ^ consists of separate

1 Compare footnote to page 116 (Tr.).

106 DEDUCTIONS FROM ELECTRODYNAMICS

terms, every one of which corresponds to a separate ordinal number n and to a simple periodic partial vibration. Strictly speaking this sum does not represent a continuous sequence of frequencies v, since n is an integral number. But n is, according to (150), so enormously large for all frequencies which need be considered that the frequencies v corresponding to the successive values of n lie very close together. Hence the interval dv, though infinitesimal compared with v, still contains a large number of partial vibrations, say n', where

dv = ^ (155)

If now in (154) we equate, instead of the total energy densities, the energy densities corresponding to the interval dv only, which are independent of those of the other spectral regions, we obtain

or, according to (155),

3T

(156)

where we denote by Cn2 the average value of Cn2 in the interval from n to n+n'. The existence of such an average value, the magnitude of which is independent of n, provided n' be taken small compared with n, is, of course, not self-evident at the outset, but is due to a special property of the function Ez which is peculiar to stationary heat radiation. On the other hand, since many terms contribute to the mean value, nothing can be said either about the magnitude of a separate term Cn2, or about the connection of two consecutive terms, but they are to be regarded as perfectly independent of each other.

In a very similar manner, by making use of (24), we find for the specific intensity of a monochromatic plane polarized ray, travelling in any direction whatever,

^~^- Ciw

64?r2

STATIONARY FIELD OF RADIATION 107

From this it is apparent, among other things, that, according to the electromagnetic theory of radiation, a monochromatic light or heat ray is represented, not by a simple periodic wave, but by a superposition of a large number of simple periodic waves, the mean value of which constitutes the intensity of the ray. In accord with this is the fact, known from optics, that two rays of the same color and intensity but of different origin never interfere with each other, as they would, of necessity, if every ray were a simple periodic one.

Finally we shall also perform the spectral resolution of the mean value of Ez2, by writing

00

dv (158)

Then by comparison with (151), (154), and (156) we find

J,-f «. -£? (159)

O fj

According to (157), Jv is related to Ky, the specific intensity of radiation of a plane polarized ray, as follows:

(160)

111. Black radiation is frequently said to consist of a large number of regular periodic vibrations. This method of expres sion is perfectly justified, inasmuch as it refers to the resolution of the total vibration in a Fourier's series, according to equation (149), and often is exceedingly well adapted for convenience and clearness of discussion. It should, however, not mislead us into believing that such a " regularity" is caused by a special physical property of the elementary processes of vibration. For the resolvability into a Fourier's series is mathematically self-evident and hence, in a physical sense, tells us nothing new. In fact, it is even always possible to regard a vibration which is damped to an arbitrary extent as consisting of a sum of regular periodic partial vibrations with constant amplitudes and constant phases. On the contrary, it may just as correctly be said that in airnature there is no process more complicated than the vibrations of black

108 DEDUCTIONS FROM ELECTRODYNAMICS

radiation. In particular, these vibrations do not depend in any characteristic manner on the special processes that take place in the centers of emission of the rays, say on the period or the damping of the emitting particles; for the normal spectrum is distinguished from all other spectra by the very fact that all individual differences caused by the special nature of the emitting substances are perfectly equalized and effaced. Therefore to attempt to draw conclusions concerning the special properties of the particles emitting the rays from the elementary vibra tions in the rays of the normal spectrum would be a hopeless undertaking.

In fact, black radiation may just as well be regarded as con sisting, not of regular periodic vibrations, but of absolutely irregular separate impulses. The special regularities, which we observe in monochromatic light resolved spectrally, are caused merely by the special properties of the spectral apparatus used, e.g., the dispersing prism (natural periods of the molecules), or the diffraction grating (width of the slits) . % Hence it is also in correct to find a characteristic difference between light rays and Roentgen rays (the latter assumed as an electromagnetic process in a vacuum) in the circumstance that in the former the vibra tions take place with greater regularity. Roentgen rays may, under certain conditions, possess more selective properties than light rays. The resolvability into a Fourier's series of partial vibrations with constant amplitudes and constant phases exists for both kinds of rays in precisely the same manner. What especially distinguishes light vibrations from Roentgen vibrations is the much smaller frequency of the partial vibrations of the former. To this is due the possibility of their spectral resolution, and probably also the far greater regularity of the changes of the radiation intensity in every region of the spectrum in the course of time, which, however, is not caused by a special property of the elementary processes of vibration, but merely by the constancy of the mean values.

112. The elementary processes of radiation exhibit regularities only when the vibrations are restricted to a narrow spectral region, that is to say in the case of spectroscopically resolved light, and especially in the case of the natural spectral lines. If, e.g., the amplitudes Cn of the Fourier's series (149) differ from zero only

STATIONARY FIELD OF RADIATION 109

between the ordinal numbers n = no and n = ni, where -

UQ

is small, we may write

(161) where

m

Co COS 0o = Xj ^» COS ( r — ~~ ~~ 0'

•^^^ \ I

no ni

Co sin 0o = - Cn sin

n-no)t \ p- -- B J

and E2 may be regarded as a single approximately periodic vibra tion of frequency vo = — with an amplitude Co and a phase-

constant 0o which vary slowly and irregularly.

The smaller the spectral region, and accordingly the smaller

— , the slower are the fluctuations ("Schwankungen") of

fto

Co and &o, and the more regular is the resulting vibration and also the larger is the difference of path for which radiation can inter fere with itself. If a spectral line were absolutely sharp, the radiation would have the property of being capable of interfering with itself for differences of path of any size whatever. This case, however, according to Sec. 18, is an ideal abstraction, never occurring in reality.

PART III ENTROPY AND PROBABILITY

CHAPTER I

FUNDAMENTAL DEFINITIONS AND LAWS. HYPOTHESIS OF QUANTA

113. Since a wholly new element, entirely unrelated to the fundamental principles of electrodynamics, enters into the range of investigation with the introduction of probability considera tions into the electrodynamic theory of heat radiation, the ques tion arises at the outset, whether such considerations are justi fiable and necessary. At first sight we might, in fact, be inclined to think that in a purely electrodynamical theory there would be no room at all for probability calculations. For since, as is well known, the electrodynamic equations of the field together with the initial and boundary conditions determine uniquely the way in which an electrodynamical process takes place, in the course of time, considerations which lie outside of the equations of the field would seem, theoretically speaking, to be uncalled for and in any case dispensable. For either they lead to the same results as the fundamental equations of electrodynamics and then they are superfluous, or they lead to different results and in this case they are wrong.

In spite of this apparently unavoidable dilemma, there is a flaw in the reasoning. For on closer consideration it is seen that what is understood in electrodynamics by " initial and boundary" conditions, as well as by the "way in which a process takes place in the course of time," is entirely different from what is denoted by the same words in thermodynamics. In order to make this evident, let us consider the case of radiation in vacua, uniform in all directions, which was treated in the last chapter.

From the standpoint of thermodynamics the state of radiation is completely determined, when the intensity of monochromatic radiation K, is given for all frequencies, v. The electrodynamical observer, however, has gained very little by this single statement; because for him a knowledge of the state requires that every one

8 113

114 ENTROPY AND PROBABILITY

of the six components of the electric and magnetic field-strength be given at all points of the space; and, while from the thermo- dynamic point of view the question as to the way in which the process takes place in time is settled by the constancy of the intensity of radiation Kv, from the electrodynamical point of view it would be necessary to know the six components of the field at every point as functions of the time, and hence the ampli tudes Cn and the phase-constants 9n of all the several partial vibrations contained in the radiation would have to be calculated. This, however, is a problem whose solution is quite impossible, for the data obtainable from the measurements are by no means sufficient. The thermodynamically measurable quan tities, looked at from the electrodynamical standpoint, represent only certain mean values, as we saw in the special case of stationary radiation in the last chapter.

We might now think that, since in thermodynamic measure ments we are always concerned with mean values only, we need consider nothing beyond these mean values, and, therefore, need not take any account of the particular values at all. This method is, however, impracticable, because frequently and that too just in the most important cases, namely, in the cases of the processes of emission and absorption, we have to deal with mean values which cannot be calculated unambiguously by electrodynamical methods from the measured mean values. For example, the mean value of Cn cannot be calculated from the mean value of Cn2, if no special information as to the particular values of Cn is available.

Thus we see that the electrodynamical state is not by any means determined by the thermodynamic data and that in cases where, according to the laws of thermodynamics and according to all experience, an unambiguous result is to be expected, a purely electrodynamical theory fails entirely, since it admits not one definite result, but an infinite number of different results.

114. Before entering on a further discussion of this fact and of the difficulty to which it leads in the electrodynamical theory of heat radiation, it may be pointed out that exactly the same case and the same difficulty are met with in the mechanical theory of heat, especially in the kinetic theory of gases. For when, for example, in the case of a gas flowing out of an opening at the time

FUNDAMENTAL DEFINITIONS AND LAWS 115

t = 0, the velocity, the density, and the temperature are given at every point, and the boundary conditions are completely known, we should expect, according to all experience, that these data would suffice for a unique determination of the way in which the process takes place in time. This, however, from a purely mechanical point of view is not the case at all; for the positions and velocities of all the separate molecules are not at all given by the visible velocity, density, and temperature of the gas, and they would have to be known exactly, if the way in which the process takes place in time had to be completely calculated from the equations of motion. In fact, it is easy to show that, with given initial values of the visible velocity, density, and tempera ture, an infinite number of entirely different processes is mechan ically possible, some of which are in direct contradiction to the principles of thermodynamics, especially the second principle.

115. From these considerations we see that, if we wish to cal culate the way in which a thermodynamic process takes place in time, such a formulation of initial and boundary conditions as is perfectly sufficient for a unique determination of the process in thermodynamics, does not suffice for the mechanical theory of heat or for the electrodynamical theory of heat radiation. On the contrary, from the standpoint of pure mechanics or electro dynamics the solutions of the problem are infinite in number. Hence, unless we wish to renounce entirely the possibility of representing the thermodynamic processes mechanically or elec- trodynamically, there remains only one way out of the difficulty, namely, to supplement the initial and boundary conditions by special hypotheses of such a nature that the mechanical or electrodynamical equations will lead to an unambiguous result in agreement with experience. As to how such an hypothesis is to be formulated, no hint can naturally be obtained from the principles of mechanics or electrodynamics, for they leave the question entirely open. Just on that account any mechanical or electrodynamical hypothesis containing some further specializa tion of the given initial and boundary conditions, which cannot be tested by direct measurement, is admissible a priori. What hypothesis is to be preferred can be decided only by testing the results to which it leads in the light of the thermodynamic prin ciples based on experience.

116 ENTROPY AND PROBABILITY

116. Although, according to the statement just made, a deci sive test of the different admissible hypotheses can be made only a posteriori, it is nevertheless worth while noticing that it is possi ble to obtain a priori, without relying in any way on thermody namics, a definite hint as to the nature of an admissible hypothesis. Let us again consider a flowing gas as an illustration (Sec. 114). The mechanical state of all the separate gas molecules is not at all completely defined by the thermodynamic state of the gas, as has previously been pointed out. If, however, we consider all conceivable positions and velocities of the separate gas molecules, consistent with the given values of the visible velocity, density, and temperature, and calculate for every combination of them the mechanical process, assuming some simple law for the impact of two molecules, we shall arrive at processes, the vast majority of which agree completely in the mean values, though perhaps not in all details. Those cases, on the other hand, which show appreciable deviations, are vanishingly few, and only occur when certain very special and far-reaching conditions between the coordinates and velocity-components of the molecules are satisfied. Hence, if the assumption be made that such special conditions do not exist, however different the mechanical details may be in other respects, a form of flow of gas will be found, which may be called quite definite with respect to all measurable mean values — and they are the only ones which can be tested experimentally — although it will not, of course, be quite definite in all details. And the remarkable feature of this is that it is just the motion obtained in this manner that satisfies the postu lates of the second principle of thermodynamics.

117. From these considerations it is evident that the hypothe ses whose introduction was proven above to be necessary com pletely answer their purpose, if they state nothing more than that exceptional cases, corresponding to special conditions which exist between the separate quantities determining the state and which cannot be tested directly, do not occur in nature. In mechanics this is done by the hypothesis1 that the heat motion is a " molecu lar chaos";2 in electrodynamics the same thing is accomplished

1L. Boltzrnann, Vorlesungen uber Gastheorie 1, p. 21, 1896. Wiener Sitzungsberichte 78, Juni, 1878, at the end. Compare also S. H. Burbury, Nature, 51, p. 78, 1894.

2 Hereafter Boltzmann'a "Unordnung" will be rendered by chaos, "ungeordnet" by chaotic (Tr.).

FUNDAMENTAL DEFINITIONS AND LAWS 117

by the hypothesis of ''natural radiation/' which states that there exist between the numerous different partial vibrations (149) of a ray no other relations than those caused by the measurable mean values (compare below, Sec. 148). If, for brevity, we denote any condition or process for which such an hypothesis holds as an " elemental chaos," the principle, that in nature any state or any process containing numerous elements not in themselves measurable is an elemental chaos, furnishes the necessary condition for a unique determination of the measurable processes in mechan ics as well as in electrodynamics and also for the validity of the second principle of thermodynamics. This must also serve as a mechanical or electrodynamical explanation of the conception of entropy, which is characteristic of the second law and of the closely allied concept of temperature.1 It also follows from this that the significance of entropy and temperature is, according to their nature, connected with the condition of an elemental chaos. The terms entropy and temperature do not apply to a purely periodic, perfectly plane wave, since all the quantities in such a wave are in themselves measurable, and hence cannot be an elemental chaos any more than a single rigid atom in motion ran. The necessary condition for the hypothesis of an elemental chaos and with it for the existence of entropy and tempera ture can consist only in the irregular simultaneous effect of very many partial vibrations of different periods, which are propagated in the different directions in space independent of one another, or in the irregular flight of a multitude of atoms.

118. But what mechanical or electrodynamical quantity represents the entropy of a state? It is evident that this quan tity depends in some way on the ''probability" of the state. For since an elemental chaos and the absence of a record of any individual element forms an essential feature of entropy, the tendency to neutralize any existing temperature differences, which is connected with an increase of entropy, can mean nothing for the mechanical or electrodynamical observer but that uniform

i To avoid misunderstanding I must emphasize that the question, whether the hypothesis of elemental chaos is really everywhere satisfied in nature, is not touched upon by the pre ceding considerations. I intended only to show at this point that, wherever this hypothesis does not hold, the natural processes, if viewed from the thermodynamic (macroscopic) point of view, do not take place unambiguously.

118 ENTROPY AND PROBABILITY

distribution of elements in a chaotic state is more probable than any other distribution.

Now since the concept of entropy as well as the second prin ciple of thermodynamics are of universal application, and since on the other hand the laws of probability have no less universal validity, it is to be expected that the connection between entropy and probability should be very close. Hence we make the following proposition the foundation of our further discussion: The entropy of a physical system in a definite state depends solely on the probability of this state. The fertility of this law will be seen later in several cases. We shall not, however, attempt to give a strict general proof of it at this point. In fact, such an attempt evidently would have no meaning at this point. For, so long as the "probability" of a state is not numerically defined, the correctness of the proposition cannot be quantitatively tested. One might, in fact, suspect at first sight that on this account the proposition has no definite physical meaning. It may, however, be shown by a simple deduction that it is possible by means of this fundamental proposition to determine quite generally the way in which entropy depends on probability, without any further discussion of the probability of a state.

119. For let S be the entropy, W the probability of a physical system in a definite state; then the propositon states that

S=f(W) (162)

where /(TF) represents a universal function of the argument IT. In whatever way W may be defined, it can be safely inferred from the mathematical concept of probability that the probability of a system which consists of two entirely independent1 systems is equal to the product of the probabilities of these two systems separately. If we think, e.g., of the first system as any body whatever on the earth and of the second system as a cavity con taining radiation on Sirius, then the probability that the terres trial body be in a certain state 1 and that simultaneously the radiation in the cavity in a definite state 2 is

W = W1Wt, (163)

1 It is well known that the condition that the two systems be independent of each other is essential for the validity of the expression (163) . That it is also a necessary condition for the additive combination of the entropy was proven first by M. Laue in the case of optically coherent rays. Annalen d. Physik, 20, p. 365, 1906.

FUNDAMENTAL DEFINITIONS AND LAWS 119

where Wi and W 2 are the probabilities that the systems involved are in the states in question.

If now Si and $2 are the entropies of the separate systems in the two states, then, according to (162), we have

But, according to the second principle of thermodynamics, the total entropy of the two systems, which are independent (see footnote to preceding page) of each other, is S = $i+£2 and hence from (162) and (163)

From this functional equation / can be determined. For on differentiating both sides with respect to Wi, TF2 remaining con stant, we obtain

On further differentiating with respect to TP2, Wi now remaining constant, we get

or

The general integral of this differential equation of the second order is

/(Tf)=/clog TF+ const. Hence from (162) we get

S = k log W-- const.,

an equation which determines the general way in which the en tropy depends on the probability. The universal constant of integration k is the same for a terrestrial as for a cosmic system, and its value, having been determined for the former, will remain valid for the latter. The second additive constant of integration may, without any restriction as regards generality, be included as a constant multiplier in the quantity W, which here has not yet been completely denned, so that the equation reduces to

S = klogW. C^

120. The logarithmic connection between entropy and prob ability was first stated by L. Boltzmann1 in his kinetic theory of

1 L. Boltzmann, Vorlesungen iiber Gastheorie, 1, Sec. 6.

120 ENTROPY AND PROBABILITY

gases. Nevertheless our equation (164) differs in its meaning from the corresponding one of Boltzmann in two essential points.

Firstly, Boltzmann1 s equation lacks the factor k, which is due to the fact that Boltzmann always used gram-molecules, not the molecules themselves, in his calculations. Secondly, and this is of greater consequence, Boltzmann leaves an additive constant undetermined in the entropy S as is done in the whole of classical thermodynamics, and accordingly there is a constant factor of proportionality, which remains undetermined in the value of the probability W.

In contrast with this we assign a definite absolute value to the entropy S. This is a step of fundamental importance, which can be justified only by its consequences. As we shall see later, this step leads necessarily to the " hypothesis of quanta" and moreover it also leads, as regards radiant heat, to a definite law of distribution of energy of black radiation, and, as regards heat energy of bodies, to Nernst's heat theorem.

From (164) it follows that with the entropy S the probability W is, of course, also determined in the absolute sense. We shall designate the quantity W thus defined as the " thermodynamic probability," in contrast to the " mathematical probability," to which it is proportional but not equal. For, while the mathe matical probability is a proper fraction, the thermodynamic probability is, as we shall see, always an integer.

121. The relation (164) contains a general method for calcu lating the entropy S by probability considerations. This, however, is of no practical value, unless the thermodynamic probability W of a system in a given state can be expressed numerically. The problem of finding the most general and most precise definition of this quantity is among the most important problems in the mechanical or electrodynamical theory of heat. It makes it necessary to discuss more fully what we mean by the "state" of a physical system.

By the state of a physical system at a certain time we mean the aggregate of all those mutually independent quantities, which determine uniquely the way in which the processes in the system take place in the course of time for given boundary conditions. Hence a knowledge of the state is precisely equivalent to a knowl edge of the "initial conditions." If we now take into account

FUNDAMENTAL DEFINITIONS AND LAWS 121

the considerations stated above in Sec. 113, it is evident that we must distinguish in the theoretical treatment two entirely differ ent kinds of states, which we may denote as " microscopic" and " macroscopic" states. The microscopic state is the state as described by a mechanical or electrodynamical observer; it con tains the separate values of all coordinates, velocities, and field- strengths. The microscopic processes, according to the laws of mechanics and electrodynamics, take place in a perfectly unam biguous way; for them entropy and the second principle of ther modynamics have no significance. The macroscopic state, however, is the state as observed by a thermodynamic observer; any macroscopic state contains a large number of microscopic ones, which it unites in a mean value. Macroscopic processes take place in an unambiguous way in the sense of the second principle, when, and only when, the hypothesis of the elemental chaos (Sec. 117) is satisfied.

122. If now the calculation of the probability W of a state is in question, it is evident that the state is to be thought of in the macroscopic sense. The first and most important question is now: How is a macroscopic state defined? An answer to it will dispose of the main features of the whole problem.

For the sake of simplicity, let us first consider a special case, that of a very large number, N, of simple similar molecules. Let the problem be solely the distribution of these molecules in space within a given volume, V, irrespective of their velocities, and fur ther the definition of a certain macroscopic distribution in space. The latter cannot consist of a statement of the coordinates of all the separate molecules, for that would be a definite microscopic distribution. We must, on the contrary, leave the positions of the molecules undetermined to a certain extent, and that can be done only by thinking of the whole volume V as bein^ divided into a number of small but finite space elements, G, each contain ing a specified number of molecules. By any such statement a definite macroscopic distribution in space is defined. The man ner in which the molecules are distributed within every separate space element is immaterial, for here the hypothesis of elemental chaos (Sec. 117) provides a supplement, which insures the unam- biguity of the macroscopic state, in spite of the microscopic indefiniteness. If we distinguish the space elements in order by

122 ENTROPY AND PROBABILITY

the numbers 1, 2, 3, ..... and, for any particular macro scopic distribution in space, denote the number of the molecules lying in the separate space elements by Ni, #2, #3 ..... , then to every definite system of values Ni, #2, #3 ..... , there corresponds a definite macroscopic distribution in space. We have of course always:

#i+#2+#3+ ..... =N (165)

or if

=1. (167)

The quantity Wi may be called the density of distribution of the molecules, or the mathematical probability that any molecule selected at random lies in the ith space element.

If we now had, e.g., only 10 molecules and 7 space elements, a definite space distribution would be represented by the values:

#1 = 1, #2 = 2, N3 = 0, #4 = 0, #5 = 1, #6 = 4, #7 = 2, (168)

which state that in the seven space elements there lie respectively 1, 2, 0, 0, 1, 4, 2 molecules.

123. The definition of a macroscopic distribution in space may now be followed immediately by that of its thermodynamic probability W. The latter is founded on the consideration that a certain distribution in space may be realized in many different ways, namely, by many different individual coordinations or " complexions," according as a certain molecule considered will happen to lie in one or the other space element. For, with a given distribution of space, it is of consequence only how many, not which, molecules lie in every space element.

The number of all complexions which are possible with a given distribution in space we equate to the thermodynamic probability W of the space distribution.

In order to form a definite conception of a certain complexion, we can give the molecules numbers, write these numbers in order from 1 to #, and place below the number of every molecule the number of that space element to which the molecule in ques tion belongs in that particular complexion. Thus the following

FUNDAMENTAL DEFINITIONS AND LAWS 123

table represents one particular complexion, selected at random, for the distribution in the preceding illustration

123456789 10 f .

617562266 7

By this the fact is exhibited that the Molecule 2 lies in space element 1. Molecules 6 and 7 lie in space element 2. Molecule 4 lies in space element 5. Molecules 1, 5, 8, and 9 lie in space element 6. Molecules 3 and 10 lie in space element 7.

As becomes evident on comparison with (168), this com plexion does, in fact, correspond in every respect to the space distribution given above, and in .a similar manner it is easy to exhibit many other complexions, which also belong to the same space distribution. The number of all possible complexions required is now easily found by inspecting the lower of the two lines of figures in (169). For, since the number of the molecules is given, this line of figures contains a definite number of places. Since, moreover, the distribution in space is also given, the num ber of times that every figure (i.e., every space element) appears in the line is equal to the number of molecules which lie in that particular space element. But every change in the table gives a new particular coordination between molecules and space elements and hence a new complexion. Hence the number of the possible complexions, or the thermodynamic probability, W, of the given space distribution, is equal to the number of ''per mutations with repetition" possible under the given conditions. In the simple numerical example chosen, we get for W, according to a well-known formula, the expression

1!2!0!0!1!4!2!

The form of this expression is so chosen that it may be applied easily to the general case. The numerator is equal to factorial N, N being the total number of molecules considered, and the denominator is equal to the product of the factorials of the num bers, Ni, Nz, Ns, ..... of the molecules, which lie in every separate space element and which, in the general case, must be

124 ENTROPY AND PROBABILITY

thought of as large numbers. Hence we obtain for the required probability of the given space distribution

Since all the N's are large numbers, we may apply to their factorials Stirling's formula, which for a large number may be abridged1 to2