Walking further with Planck

Max Planck, in his 14 December 1900 presentation to the German Physical Society, reported he was able to “derive deductively an expression for the entropy of a monochromatically vibrating resonator and thus for the energy distribution in a stationary radiation state, that is, in the normal spectrum.”

 

The title of his talk was “On the Theory of the Energy Distribution Law of the Normal Spectrum.”  In more modern language, he was talking about electromagnetic standing waves or normal modes of radiation in what we now call a cavity, which contained abstract electrically charged oscillators or resonators We may imagine the cavity to be a heated metallic cube insulated from the outside but not sealed. It’s a sort of oven, actually, but instead of having a big door to open to see what’s going on inside, it has a small hole through which light to be spectrally analyzed is emitted. 

The small hole absorbs light which is then mixed in with the light already in the cavity. No light is reflected  from the hole, so it acts as a near-perfect absorber and emitter of radiation, and its spectrum is that of a black-body.

Planck informed the assembled GPS members that his theoretical analysis of black-body radiation was based on “the laws of electromagnetic radiation, thermodynamics and probability calculus.” Precise measurements were being made in 1900 of the spectrum of cavity radiation, so Planck had experimental results to compare to his theory. Fitting his calculated energy distribution to the experimental results played an important role  in his discovery.

Inside Planck’s imaginary cavity there are “monochromatically vibrating resonators,” which he imagines divided up into different groups according to their frequency of radiation, with N resonators in the group having frequency υ, N’ resonators in the group having frequency υ’, N’’ in the group having frequency υ’’, and so on.  Planck says these resonators are “at large distances apart,” and are “enclosed in a diathermic medium with light velocity c and bounded by reflecting walls.” 

He assigns a total energy E_t to the radiation and the resonators and says: “The question is how in a stationary state this energy is distributed over the vibrations of the resonator and over the various frequencies of the radiation present in the medium, and what will be the temperature of the total system.”

Planck’s prescription for the energy of the resonators is that they have “arbitrary definite energies.”  He labels the different definite energies just as he labeled the resonators and their frequencies, with an increasing number of apostrophes: E, E’, E’’, and so on.   Then he says:

 “The sum E + E’ + E’’ + E’’’ + … = E₀  must, of course, be less than E_t. The remainder E_t - E₀ pertains then to the radiation present in the medium. We must now give the distribution of the energy over the separate resonators of each group, first of all the distribution of the energy E over the N resonators of frequency υ.  If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways.  We consider, however—and this is the most essential point of the whole calculation—E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55 x 10^-27 erg·sec.  This constant multiplied by the common frequency υ of the resonators gives us the energy element ε, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.  If the ratio is not an integer, we take for P an integer in the neighborhood.”

What does this inexactness mean? If we take ε to be the unit of energy then E must consist of a certain, exact number of these units.  I mean, this is supposed to be where quantization occurs!  Contained in this question is the question of discrete energy levels versus continuous frequency values.  We’ll hold off on this question for the moment.

 

Planck then calculates the number of ways to distribute P energy elements over the N resonators in order to calculate the entropy of these resonators.  He is doing his “probability calculus” first, then his thermodynamics calculations. His calculations for electromagnetic radiation were done in earlier papers. The purpose of his December 19 presentation, he says, is “to explain as clearly as possible the real core of the theory.”

Regarding the number of ways to distribute P elements of energy over N resonators of frequency υ, Planck continues: “Each of these ways of distribution we call a ‘complexion,’ using an expression introduced by Mr. Boltzmann for a similar quantity.”  Then Planck gives a simple numerical example of a complexion:  He chooses P = 100 units of energy to be distributed in one particular way over N = 10 resonators.  His table of numbers is:

1	2	3	4	5	6	7	8	9	10
--------------------------------------
7	38	11	0	9	2	20	4	4	5

Regarding this table he says, “The number of all possible complexions is clearly equal to the number of all possible sets of numbers which one can obtain for the lower sequence, for given N and P.  To avoid all misunderstandings, we remark that two complexions must be considered to be different if the corresponding sequences contain the same numbers, but in different order.”

Now for the calculation. We have 10 “boxes” and 100 “balls,” and want to know how many ways there are to distribute the 100 balls in the 10 boxes. One way to do this calculation is to show Planck’s table of numbers in a pictorial fashion like so

|°°°°°°°|°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°|°°°°°°°°°°°||°°°°°°°°°|°°|°°°°°°°°°°°°°°°°°°°°|°°°°|°°°°|°°°°°|

We then imagine the ways that the different symbols shown here can be redistributed.  According to Bernard Lavenda (Statistical Physics: A Probablistic Approach, 1991, p. 100) this combinatorial idea was first expressed as a "joke" in an October 1914 letter from Kamerlingh Onnes and Paul Ehrenfest to H. A. Lorentz.  But it works quite well, which is one reason it can be considered funny.  Archimedes was probably laughing when he ran through the streets naked shouting "Eureka!"  In contrast, Planck in 1900 merely wrote down the general formula (see below), and said it came from “the theory of permutations.”

We take the symbols ° and | as items to be redistributed. Thus we have balls and “walls” rather than balls and boxes.  The key idea is that the walls on the ends don’t get redistributed, since we then would have some balls outside the boxes.  So we leave those two walls out of the calculation, and count the number of redistributions we can make of the items inside those two walls.  What we have is one long box with 100 balls and 9 walls (count ‘em yourself) inside it. So we have 9 identical items of one type and 100 identical items of another type. In the general case, we have (N-1) walls and P balls. Keep this in mind when you look at the general formula below!

To get to the general formula, we need to start with a simpler situation.  If we had 109 items and none were identical to each other—let’s say you typed in 109 unique keyboard characters—how many ways to redistribute these would there be? There would be 109! ways.

(The exclamation mark is factorial notation. Look it up, or try it for yourself with these four words: Bridge Ices Before Road.  There are 4! = 4·3·2·1 = 24 ways to distribute these four non-identical words. To show this, start by writing the 4 words in a column, then put the combinations of the remaining 3 words in a column to the right of each of these, and so on for the 2 remaining words and then the one remaining word.)

Now we go to the next level of complexity.  If 9 of these 109 are identical we would have fewer distinguishable ways of redistributing the 109 items. There are 9! indistinguishable ways of redistributing 9 identical items, and although it seems pointless to consider these indistinguishable configurations, we have to divide 109! by 9! to get the number of distinguishable combinations when we have 9 identical items among the 109.

In our case we also have a group of 100 items out of the 109 that are identical. So we have to divide 109! by 100! and by 9!,  giving a total of 

109!/(100!)(9!)  = 109·108·107·106·105·104·103·102·101·100!/(100!)(9!) 

= 109·108·107·106·105·104·103·102·101/9! = 4263421511271, a very big number on the order of 10¹², or a terabyte of data--well, okay, not the number itself. If you had to store every number between 0 and 4263421511271 that would be about a terabyte.

The general expression for the number of ways P identical energy elements can be divided up among N resonators is

[P + (N-1)]!/[P! (N-1)!], 

which Planck writes as

(N+P-1)!

------------

(N-1)! P!

 But remember we have N’, N’’, etc. and P’, P’’, etc.  Planck says:

We perform the same calculation for the resonators of the other groups, by determining for each group of resonators the number of possible complexions for the energy given to that group.  The multiplication of all numbers obtained this way gives us then the total number R of all possible complexions for the arbitrary assigned energy distribution over all resonators.

In the same way any other arbitrarily chosen energy distribution E, E’, E’’,… will correspond to a definite number R of all possible complexions, and R is evaluated in the above manner.  Among all energy distributions which are possible for a constant E₀ = E + E’ + E’’ + E’’’ + … there is one well-defined one for which the number of possible complexions R₀ is larger than for any other distribution.  We look for this distribution, if necessary by trial, since this will just be the distribution taken up by the resonators in the stationary radiation field if the resonators together possess the energy E₀.  These quantities E, E’, E’’,… can then be expressed in terms of E₀.  Dividing E by N, E’ by N’, and so on, we obtain the stationary value [the average value] of the energy U_υ , U’_υ’ , U’’_υ’’ , … of a single resonator of each group, and thus the spatial density of the corresponding radiation energy in a diathermic medium in the spectral range υ to υ+dυ,

u_υdυ = (8πυ²/c³) U_υ dυ,

so that the energy of the medium is also determined.

At this point we need to pause and be reminded of what Max Karl Ernst Ludwig (Planck) means when he says “normal spectrum” or “normal energy distribution.”  What he says is, “The normal energy distribution is then the one in which the radiation densities of all different frequencies have the same temperature.” 

The best way to understand this statement is to quote what he says just before it.  He talks about the hypothesis of “natural radiation,” and even puts those words in quotes himself, then says: “the law of energy distribution in the normal spectrum is completely determined when one succeeds in calculating the entropy S of an irradiated, monochromatic, vibrating resonator as a function of its vibrational energy U.  Since one then obtains, from the relationship dS/dU = 1/T, the dependence of the energy U on the temperature T, and since the energy is also related to the density of radiation at the corresponding frequency by a simple relation, one also obtains the dependence of this density of radiation on the temperature.” 

The “simple relation” is the one above for u_υ  in the frequency range υ to υ+dυ.  Planck had earlier derived the factor in parenthesis (Lord Rayleigh had done it even earlier that same year). This factor is what makes the makes the spectrum "normal," and it represents the number of resonators or oscillators (or modes, in modern language) in the given frequency range.  Planck’s big deal was that he found the expression for the average energy U_υ that fit the data for black-body cavity radiation.  So we’ll be coming back to that derivation next. 

In the meantime, here’s a quote from Peter Milonni’s book The Quantum Vacuum concerning how Planck used the permutation formula above:  “Planck counted the number of ways, or ‘complexions,’ over which P energy elements could be distributed among N radiators. His counting procedure was totally at odds with classical statistical methods in its treatment of the energy elements as fundamentally indistinguishable.  In one sense Planck was following Boltzmann in regarding all complexions equally likely, but of course his way of counting the number of complexions was radically different.  His ‘energy elements’ obeyed what would much later be recognized as Bose-Einstein statistics.”