More walking with Max Karl Ernst Ludwig Planck

Coincidentally, who does Planck share one or more of his names with? Max Born and Max Theodor Felix von Laue.  Max von Sydow. Peter Max.  Karl Marx.  Karl Childers (“Some folks call it a sling blade, I call it a Kaiser blade.”). Ernst Mach.  Max Ernst.  Laura Kathleen Planck.  And Ludwig Boltzmann ...

Time now to get back to Planck, Boltzmann and Einstein on the related subjects of entropy, combinatorics and quanta. But first, more about names: Boltzmann had the usual first, middle and last names: Ludwig Eduard Boltzmann; Einstein had just a first and last name, no middle name.  A curious (unrelated) fact is that Boltzmann’s younger brother, who was born in 1846 and died of pneumonia in his early teens, was named Albert, and Albert Einstein’s younger son, born in 1910, four years after Ludwig Boltzmann’s suicide, was named Eduard; he was diagnosed in his early 20s as schizophrenic and spent most of his life in Burghölzli sanitarium in Zurich, dying at age 55 (he also had respiratory problems when he was seven, in 1917, and spent time in the tuberculosis sanatorium in Arosa, near Davos, made famous in Thomas Mann's 1924 novel Der Zauberburg [The Magic Mountain]).

And secondly, before getting to the calculations of Boltzmann, Planck, and Einstein, let’s hear from Herb Callen on the meaning of entropy. Planck’s decision to calculate the entropy of his oscillator model led him to the discovery of the quantum of action in 1900. The quotes below are from Callen’s 1985 book Thermodynamics and an Introduction to Thermostatistics, which I’m glad Mark Loewe chose to use as the text for his course in Statistical Mechanics I took at Southwest Texas State University (now called Texas State University-San Marcos) in 1995.

Entropy as the central concept in thermodynamics, p. 329: Thermodynamics constitutes a powerful formalism of great generality, erected on a basis of a very few, very simple hypotheses. The central concept introduced through those hypotheses is the entropy. It enters the formalism abstractly as the variational function in a mathematical extremum principle determining equilibrium states. In the resultant formalism, however, the entropy is one of a set of extensive parameters, together with the energy, volume, mole numbers and magnetic moment. As these latter quantities each have clear and fundamental physical interpretations it would be strange indeed if the entropy alone were to be exempt from physical interpretation.

Entropy, statistical mechanics, and model systems, p. 333: In the eleven chapters of this book devoted to thermodynamic theory there were few references to specific model systems, and those occasional references were kept carefully distinct from the logical flow of the general theory. In statistical mechanics we almost immediately introduce a model system, and this will be followed by a considerable number of others. The difference is partially a matter of convention. To some extent it reflects the simplicity of the general formalism of statistical mechanics, which merely adds the logical interpretation of the entropy to the formalism of thermodynamics; the interest therefore shifts to applications of that formalism, which underlies the various material sciences (such as solid state physics, the theory of liquids, polymer physics, and the like).But, most important, it reflects the fact that counting the number of states available to physical systems requires computational skills and experience that can be developed only by explicit application to concrete problems.

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Okay then!  Let's get back to Planck's concrete problem of electrically charged oscillators interacting with radiation inside a small, oven-like enclosure—a model system if there ever was one!

Planck’s method of counting states, calculating entropy, and then calculating the average energy of his oscillators was revolutionary, and was also confusing to his contemporaries, among them 21-year-old Albert Einstein.  Like some of Einstein’s own papers written later, the results were correct and would stand the test of time, but how the results were derived—especially Planck’s combinatorial “definition” of probability—left something to be desired. 

Boltzmann and Planck: Complexions and their Permutations

I want to look at what Planck mentioned but didn’t calculate in his December 1900 presentation and paper, in which he says he’s trying to “explain as clearly as possible the real core of the theory.” The real core of the theory is partly taken from Boltzmann's 1877 paper, where Boltzmann uses a division of molecular kinetic energies into integer multiples of a tiny, constant amount of energy he labeled as ε. Thus Boltzmann allows energy exchanges between molecules to occur only in discrete packets having values of ε,  2ε, 3ε, and so on. Planck uses ε also, but postulates that ε = hν, which is the other part of "the real core of the theory".  It’s important to keep in mind, however, that Boltzmann has only one ε, which we can call a discrete energy element, while Planck has an unlimited number of discrete energy elements whose size is directly proportional to is equal to a constant times their associated frequency.

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By the way, writing the quantization condition as ε = hν is not the mathematically correct thing to do, although just about everybody does it. Planck did it too, but he first declared both ε and ν to be singular values selected from the continuum of ε' s and ν' s.  The only problem is, he doesn't then introduce the other ε' s for the other frequencies that he labels ν', ν'', ν''', etc. (See the section below titled "Oscillators and radiation at many frequencies" where I discuss ε', ε'', ε''', etc. in their proper energy-distribution context.)

The equation ε = hν is not a quantization condition because it shows a direct proportionality between ε and ν, the same as Ohm's Law, V = RI, shows a direct proportionality between voltage and current (R=constant resistance), or Newton's 2nd law, F = ma, shows a direct proportionality between net force and acceleration (m=constant mass). And this is clearly the wrong idea for showing quantized energy levels.

Quantization, to be precise, is shown by a uniform separation of energy levels, Δε = hν₀, of a Planck oscillator having the SINGLE frequency ν₀, so the oscillator energy levels (neglecting the later-discovered zero-point energy) are 0, ε, 2ε, 3ε, … = 0, hν₀, 2hν₀, 3hν₀, ... , etc.  But ν itself is a continuous quantity in black-body radiation, so you have to be careful when thinking about Δε.  When all frequencies are considered, the Δε’s merge into a continuum of energies. Planck managed to skirt this difficulty by using his chosen notation of multiple primes, ', '', ''', etc, which wasn't a discrete notation, exactly, and wasn't a continuous notation using differentials, either.

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Let’s deal with Planck’s analysis, or lack of analysis, first, before discussing what Boltzmann did.  Planck wants to calculate the entropy of his system of electrically charged oscillators, but at first doesn’t write an expression or use a variable for counting the number of states available to this system.  Instead, he gives a particular numerical example in a table based on N = 10 oscillators and P = 100 discrete identical electromagnetic energy elements ε. The ten oscillators and the electromagnetic radiation they emit and absorb are all considered to have the same frequency, ν, singled out from the presumed continuum of frequencies in the cavity (that should raise your eyebrows).  In Planck's table, oscillator labels are on the top row, and the number of energy elements they each contain are on the bottom:

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

This sort of freshman physics numerical example is one of my favorite things Planck does in his paper: a simple, concrete example of what he calls a "complexion," a term he says Boltzmann uses for “a similar quantity.” Unfortunately, this is all Planck says in regard to Boltzmann’s 1877 analysis.  He should have said more! Even though Planck’s table is helpful, it could have been much more helpful if he’d related it numerically to Boltzmann’s complexions and permutations. 

What Planck does after showing the table only obscures the basic idea needed for counting complexions. 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.” If he’d first calculated the number of complexions for his table, he would have made the best use of it and avoided a lot of confusion among his future readers, of whom there turned out to be many. Instead he shows the expression for the number of all possible complexions for a given N and P,

(P + N – 1)! / [P! (N – 1)!],

which is the first actual formula in his paper (not counting his simple sum of the oscillator energies at different frequencies).  By not showing the Boltzmann permutation formula explicitly for the table, Planck makes it look like this is his only permutation formula, and it’s not the same as Boltzmann’s permutation formula, although Planck has so far done the same thing Boltzmann did, because he’s only looking at oscillator energy distributions at one frequency, so he's only introduced the one energy element ε.  (He has mentioned ε = hν but hasn’t used it yet.)  

Anyone familiar with the first few pages of Boltzmann’s long 1877 paper would know Planck’s “all possible complexions” formula is also Boltzmann’s formula for the sum of permutations, or, yes, for all possible complexions. (An apparent contradiction to look at later: this formula gives the number of coefficients in a multinomial expansion, not the sum of the coefficients.)  Boltzmann and Planck both use the formula in the same way, but Boltzmann first shows the all-important permutation formula, and Planck doesn’t show it at all!  (And I haven’t shown it yet, but am getting close.)

Boltzmann only needed to used this “all possible complexions” formula as the necessary normalization factor (the overall divisor) in his sum of all the different possible molecular energy permutations. This is a sum of fractions, where each fraction represents the probability of a particular “state distribution,” and this sum must equal one. Planck, on the other hand, in order to find the expression he knows he needs for the oscillators’ entropy, uses the above formula itself as a probability (he makes this declaration in his next paper after his Dec 1900 one, so I won’t go into that just yet).

Going back to Planck’s table, the one comment he makes in regard to it—but only after he’s given the above factorial formula, which doesn’t apply directly to the table—is, "two complexions must be considered to be different if the corresponding sequences contain the same numbers, but in different order.”  In other words, if we permute (exchange, rearrange, change the order of) any numbers in the bottom row of the table, we have a different complexion from the one we started with—except when we exchange the two 4’s. Planck doesn’t mention this detail about the two 4’s when he says “the theory of permutations” describes how to calculate the number of “all possible complexions.” But at least we can be thankful he included in his table two numbers that are the same, so the primary idea of what a permutation calculation is can be shown. Which is what I’ll do now.

The permutation calculation Planck didn’t do

In Planck’s table there are eight numbers that are different and two that are the same. (I’m referring to the bottom row. The top row is not even necessary, because we're looking at the different "line-ups" of the bottom row.)  If all ten of Planck’s energy element numbers were different, there would be 10! different arrangements (line-ups) of them, or 10! permutations. But two of the numbers are the same, so we have to divide by 2!, which gives 

10! / 2!  =  3,628,800 / 2  =  1,814,400.

By doing this division, we're knocking out of the permutation total the number of extra complexions there'd be if these two numbers weren’t  the same.  The more energy elements that are the same, the fewer permutations there are, so this calculation weeds out the indistinguishable combinations. Planck’s use of only two identical energy elements is actually a good way to see this kind of calculation for the first time.  Was he aware of this particular pedagogical nature of this table?

Here are a couple of other pedagogical examples, based on Planck’s N=10 and P=100:  (1) If each oscillator held 10 energy elements, how many possible complexions would there be?  Answer: 10!/10! = 1. One possible complexion, which is the same as saying one permutation (although you might wrongly want to say zero permutations).   (2) If one oscillator held all 100 energy elements, how many permutations would there be? Answer: We have nine occupation numbers that are the same, the nine zeros, so we have 10!/9! = 10 possible complexions. These are achieved by assigning the l00 energy elements in turn to each of the 10 oscillators.  

Now, let’s look at Boltzmann’s analysis and compare it to Planck’s. Boltzmann analyzed an insulated box (an oven, again!) of N gas molecules possessing and exchanging translational kinetic energies in integer multiples of ε, with the highest possible kinetic energy level of the molecules being pε. (Only one molecule could be in this p-energy level and all the rest would then have to be in the zero-energy level.  An example of this is the Planck-table example with 100 energy elements in one of the 10 oscillators. In energy-level language, this is one oscillator in the 100^th energy level.)  Thus, Boltzmann used the opposite form of assignment from what Planck later used when it comes to energies possessed by molecules. He assigned numbers of molecules to energies, or really to energy levels, rather than numbers of energy elements to molecules like Planck does for oscillators in his table.

In Boltzmann’s notation, the number of molecules in the kinetic energy level i is given by w_i. Thus, there are w₀ molecules with zero kinetic energy, w₁ molecules with one unit of kinetic energy, w₂ molecules with two kinetic energy units, and so on, up to w_p. In contrast to Planck’s energy-units-per-oscillator counting method, Boltzmann’s molecules-per-energy-level scheme automatically gives all the numbers necessary to do a permutation calculation.  The calculation of the number of complexions for ONE SET of w_i’s is

W_B = N! / (w₀! w₁! w₂ ! … w_i!… w_p!).

There are other sets of  w_i’s, all of which are determined by the constraints on energy and numbers of molecules (oscillators in Planck’s case). These constraints are given below in the “frequency-level occupation numbers” section.

Since Planck just gives a table of numbers as his example of a complexion, and then jumps to the sum-of-all-complexions formula, it’s not evident in his December 1900 paper that his complexions and Boltzmann’s are the same.  But they are precisely the same, as shown by writing out the Boltzmann permutation formula for Planck’s table of numbers. The only factor in the denominator that isn’t a 0! or a 1! is w₄ = 2! :

W_B = 10! /  (w₀! w₁! w₂! w₃ ! w₄! w₅! … w₁₀₀!)  = 10!/ (1! 0! 1! 0! 2! 1! … 0!)

= 10!/ 2! = 1,814,400.

Oscillators and radiation at many frequencies

So far, I’ve only dealt with Planck’s analysis for oscillators and radiation at one particular frequency. Now I’m going to start moving on to the case of more than one frequency, but I won’t actually get there until the next section.

Planck discussed the general case of oscillators and radiation at different frequencies in his December 1900 presentation/paper, but he didn’t show any calculations for these other frequencies. In fact, there’s a practical problem with using Planck’s notation to do the calculations for different frequencies. The problem is that he uses different numbers of accents or primes (single prime ‘, double prime ‘’, triple prime ‘’’, etc.) to indicate the different frequencies of the equilibrium radiation,

Frequencies:  ν, ν^’, ν^’’, ν^’’’, … ,

and the number of oscillators at each frequency,

Number of oscillators at different frequencies:  N + N^’ + N^’’ + N^’’’ + …  = N,

and the energy of the oscillators,

Energies possessed by these oscillators at each of these frequencies:

E + E^’ + E^’’ + E^’’’ + … =  Pε + P’ε’ + P’’ε’’ +  P’’’ε’’’  + … = E₀   .

I have to wonder why Planck didn’t use index notation instead of primes.  Was it because index notation indicates discreteness, and the frequencies of black-body radiation, as was well-known by Planck, form a continuous spectrum?  There really seems to be no in-between choice.  Either treat the frequencies, etc, as continuous quantities and use differentials and integrals, or use a counting notation such as indexes or, as in Planck’s case, superscripts that are just different numbers of freakin’ apostrophes (or accents, or primes, or whatever you want to call them).

Anyway, in Planck’s prime notation, the formula above for the total number of permutations applies only to the frequency ν.  At the frequency ν^’, which is the next higher frequency above ν, the formula a la Planck would be

(P’ + N’ – 1)! /[P’! (N’ – 1)!].

(But what would this next higher frequency ν^’ be, anyway?  Actually, it’s just a different frequency, so no need to worry about actual numbers or relative sizes for frequencies, at least not yet.  Planck didn’t worry about it, because he didn’t write down this formula with primes on it, or do any calculations for different frequencies besides ν.  But! Let’s not forget that the question of what mathematical function would fit the observed distribution of frequencies and intensities of black-body radiation at different temperatures was the central question Planck and all the others wanted to answer, and Planck did answer!)

Since Planck’s notation is worrisome, or bothersome, or just unbelievably cumbersome, I’ll use the index j to label the ν’s, N’s, E’s, and P’s for arbitrary frequencies: ν_j, N_j , E_j, and P_j .  In the corresponding Planckian Primes notation, j = 0 is no prime, j = 1 is one prime, and so on. 

Using index notation, we have a concise total-number-of-permutations formula for any frequency,

(P_j + N_j – 1)! / [P_j! (N_j – 1)!],

with  j = 0, 1, 2, 3, … .   The upper limit on j (i.e., the highest frequency) is something else we won’t worry about right now!  (Actually, Boltzmann lets his p go to infinity, which presents a problem of interpretation similar to Planck's highest frequency going to infinity.)

Now we come to the next calculation Planck didn’t do in his December 1900 paper, and also to Planck’s use of R as a variable in that paper. “The multiplication of all numbers obtained in this way,” Planck writes in regard to the factorial shown above, “gives us then the total number R of all possible complexions for the arbitrary assigned energy distribution over all resonators.” Try to imagine using Planck’s multiple-accent notation to write out such a multiplication! Maybe that’s why Planck didn’t show it. Or maybe, as I mention below, he found he didn’t really need to show the multiplication.

Using index notation, the big-boy multiplication over different frequencies is

R = П {(P_j + N_j – 1)! /[P_j! (N_j – 1)!]},

where  П  is the symbol for multiplication over all possible integer j values. Which would seem to go from one to infinity, with all the factors being large numbers, but that’s a worry for another time. (Not really a worry, we’ll just use renormalization to get rid of unwanted infinities, as in real quantum mechanics, ha. Ha!)

The final level of detail to be dealt with is the “the arbitrary assigned energy distribution over all resonators.”  This is the sum of the E_j’s, which must equal E₀, the total energy of all the oscillators.  Before getting to that, though, we need to anticipate and overcome the problem of using a factorial for further calculations, which means we need to find a usable non-factorial approximation.

This approximation is likely something Planck had in mind before writing his December 1900 paper. Indeed, it’s likely the most prominent thing he had in mind then, because he knew the form of the expression he needed. He had found it in his October 1900 paper and presentation to the German Physical Society, the paper in which he first derives the “Planck spectrum equation,” but doesn’t yet postulate ε = hν.

Also in October, he mentioned in a letter to a colleague (his friend Otto Lummer, one of the Berlin experimentalists working with cavity radiation), that he believed the entropy of a resonator should be S = α log[(β+U)^(β+U) / U^U)], where U is the resonator’s average energy and α and β are constants to be calculated from theory.  This is the form of the expression he wanted to work backwards to from the factorial expression. 

In fact, the roughest sort of approximation for very large values of n does involve just blatantly substituting nⁿ  for n!. This is a truncation of the Stirling approximation for factorials of large numbers. Using this truncation, Planck’s sum-of-all-complexions factorial becomes

(P_j + N_j – 1)! / [P_j! (N_j – 1)!]   ≈   (P_j + N_j) ^{(Pj + Nj)} /[P_j ^Pj N_j ^Nj ].

Notice that the overcounting errors introduced by using exponentials in place of factorials tend to be lessened by the fact that they’re being used in a quotient—some of the overcounting errors are “divided out.”  For example, for P=100 and N=10, here’s the comparison of the factorial quotient versus the exponential quotient:  

Factorial: (P+N-1)! / P! (N-1)! ≈ 1.44x10¹⁷⁶/(9.33x10¹⁵⁷)(362,880) = 4.25x10¹²,

Exponential: (P+N)^(P+N) / P^PN^N = 110¹¹⁰/100¹⁰⁰10¹⁰ ≈ 3.57x10²²⁴/10²¹⁰ = 3.57x10¹⁴.

Here you can see the error in the final result is only 10² out of 10¹², as compared with the individual many-factors-of-ten errors in each of the individual exponentials before doing the division.  Yeh, this approximation isn’t good for the small numbers 10 and 100.  But you get to see the actual number of “all possible complexions” for Planck’s N=10 and P=100 example.

To keep going and get to the calculation of the equilibrium entropy of the oscillators by considering more than one frequency, I now need to introduce frequency-level occupation-number counting.

Frequency-level occupation numbers

Boltzmann used a division of molecular kinetic energies into integer multiples of ε in his analysis. Since his “molecules” are considered to be structureless, with no mechanism for storing energy, they have only kinetic energy. They are definitely not considered to be electrically charged, since that would be another mechanism for losing or gaining kinetic energy in addition to the exchanges of kinetic energy in collisions

Planck’s oscillators give us just the opposite case: oscillators that are electrically charged, that don’t move, and can possess only stored energy. Also, Boltzmann’s molecules exchange energy directly with each other, whereas Planck’s oscillators don’t interact with each other. They only exchange energy with the electromagnetic fields in the cavity by absorption and emission.

But here’s a big mystery that almost nobody worries about anymore—how does an electrically charged harmonic oscillator store energy, and how can it emit and absorb discrete amounts of EM radiation?  I’ll just leave that alone right now, except to say it’s one reason I have an interest in studying Planck’s analysis. Planck himself suggested in the 1912 edition of his book The Theory of Heat Radiation that a better model than his abstract resonators might someday be found, and Einstein’s 1916-17 model, based on the discrete energy levels of Bohr’s 1913 hydrogen atom model, is indeed better than Planck’s resonators but still physically unsatisfactory.  (Yeh, to me quantum mechanics itself is physically unsatisfactory, which I guess is why I can’t quit thinking about it.)

Notice I’m calling the oscillator energy “stored energy” not “potential energy.”  The oscillators do have internal kinetic and potential energy, but these are lumped together in the oscillator amplitude, or amplitude-squared, really.  The simple harmonic oscillator energy equation is

p²/2m + ½ kx² =  ½ kA²,

which says instantaneous kinetic energy (where p is linear momentum, not our highest-energy-level p from above) plus instantaneous potential energy equals a constant, the amplitude-squared energy.  In Planck’s theory, just as in the theory of the harmonic oscillator in general, emission or absorption of energy results in a decrease or an increase, respectively, of the amplitude A in the amplitude-squared energy, ½ kA².  

Up to the point just before Planck considers oscillators of more than one frequency, Boltzmann’s and Planck’s analyses are the same. (I haven’t yet brought in Planck’s frequency-dependent quantization condition, ε_j = hυ_j. That only comes in at the very end.)  They both use discrete energy elements, which are best expressed as discrete energy levels, with energy-level occupation numbers w_i, even though Planck unfortunately doesn’t explicitly use these. In Boltzmann’s analysis, permutations of these numbers represent all the possible microscopic energy distributions. In Planck’s analysis, however, permutations of these numbers make up only a single-frequency microscopic energy distribution.

So now Planck must part ways with Boltzmann, and go all alone into infinite-dimensional frequency space. Hold that thought—it may actually mean something later! But Planck and Boltzmann do indeed part ways here, because Boltzmann in his 1877 paper goes on to do calculations for a continuum of kinetic energies, while Planck goes on to a continuum of frequencies—the continuous black-body radiation spectrum—even though he is still very discrete about it.

In the index notation I’m using, the distributions of energies at different frequencies are given by different sets of numbers {P_j }, where j = 0, 1, 2, 3, … , that enumerate how the total oscillator energy E₀ is divided up among the different oscillator frequencies, so we can call these P_j‘s the  frequency-level occupation numbers.  (They have to be multiplied by the ε_j ‘s to get energy at each frequency.  See below.)

The key to using the numbers {P_j } for calculating entropy is to note that different sets of them represent the different “arbitrary assigned energy distributions.”  Changing the arrangement of the arbitrary energies assigned to different frequencies is done by changing the P_j‘s at each frequency, and this changes R for each assigned energy distribution. Also, don’t forget that we’re looking for the equilibrium distribution of energy among oscillators and radiation at a particular temperature.  The equilibrium distribution is the one corresponding to the maximum of R with respect to changes in the P_j ‘s.

This brings us to the point at which the constraints on the total number of oscillators and total available energy need to be written down. The first constraint is that the w_i’s at any frequency must add up to N_j, the total number of oscillators at that frequency: 

w₀ + w₁ + w_{2 +} … + w_p =  Σ w_i = N_j .

Thus we have an unchanging number of oscillators at each frequency, and these must collectively change their amplitudes at different temperatures in order to maintain equilibrium with the radiation. For low temperatures, higher-frequency oscillators (higher-frequency “modes”) have zero amplitude.

The second constraint—for equilibrium at a particular temperature—is the total oscillator energy at each frequency.  In Planck’s notation, the energy elements at each frequency are given by ε, ε’, ε’’, ε’’’ …  etc.  (He only shows ε in his paper, which is a source of confusion—because of our devotion to ε = hν we tend to think it’s the only ε we need—and a sort of oversight, since he shows N, P, and E notated by multiple primes representing different frequencies.)  In the notation I’m using, the energy elements are ε_j. j = 0, 1, 2, 3,  etc.  Therefore, the total energy of the oscillators at each frequency is

E_j = Σ iε_j w_i  =  ε_j Σ iw_i  =  ε_j P_j ,

where i goes from zero to p (this is lower-case p, representing the highest energy level available at a particular frequency). We ultimately want to find the equilibrium values of the frequency-level occupation numbers P_j.

Also, I just realized, the relation of the oscillator amplitudes to the w_i’s is easy to show.  There are N_j oscillators at a particular frequency υ_j so a sum of amplitude-squared energies from one to N_j is needed:

Σ  ½ kA_i² = ½ k Σ A_i² =  ε_j P_j  = ε_j Σ iw_i .

This relation is something I’ll be working on in a later online journal entry.

Thermodynamic equilibrium occurs when the oscillators’ emission rates are equal to their absorption rates, which results when the system is held at a constant temperature for a long enough time (what determines this time, and how long is it?).  When the oscillators are in equilibrium with the radiation, the frequency-level occupation numbers can be used to tell us the intensities of different frequencies of radiation present in the black-body resonant cavity—that is, the spectrum!

This is what Planck is referring to when he mentions the total energy of the system, and says, “The question is how in a stationary state this energy is distributed over the vibrations of the resonators and over the various colors of the radiation present in the medium, and what will be the temperature of the total system.”

The undone Planck calculations leading to average oscillator energy

According to Planck’s statement about R being found by multiplication of the sum-of-the-permutations formulas for all the different frequencies, we have an unlimited product over the index j, our frequency-identifying index.  Using the approximation of factorials by exponentials, the expression is

R ≈  П  { (P_j + N_j) ^{(Pj + Nj)} /[P_j ^Pj N_j ^Nj ] } .

There’s no upper limit on the index j, because that would imply an upper limit on frequency.  But we’re going to be writing down only one representative term when we maximize the logarithm of R, because each of the j terms in the maximization expression must be zero independently of the others, due to the imposed constraint.

Maximizing R is the next thing Planck discusses but doesn’t calculate: “Among all energy distributions which are possible for a constant E₀ = E  +  E’  +  E’’  +  …  there is one well-defined one for which the number of possible complexions R₀ is larger than for any other distribution.”

At this point, Planck is preparing to calculate the entropy of the oscillators, and from that to calculate their average energy.  He doesn’t talk about using the natural log of R in the calculation, but he does say that the entropy of the system of oscillators is given by “k ln (R₀)” and that this “is the sum of the entropy of all separate resonators.”  Thus, we know the log of the R expression above is used to find the maximum value.

This is where the use of undetermined multipliers comes in, and this is the primary calculation Planck didn’t do.  The quantity log(R) is maximized subject to the restriction that the energies of the oscillators at all the various frequencies add up to E₀.  In the index notation I’m using here, the restriction is

E₀ = E₁  +  E₂  +  E₃  +  …  =  Σ_j E_j …  =  Σ_j ε_j P_j .

This equation gives us an auxiliary condition that needs to be multiplied by C, the undetermined constant, and then used to find a stationary point by variation with respect to P_j..  Here we go:

δ(log R – CE₀) =  δ( log R – CΣ_j ε_jP_j )  =  0

= δ (log П(P_j + N_j) ^{(Pj + Nj)} /[P_j ^Pj N_j ^Nj ]  – CΣ_j ε_j P_j ) 

=  δ ( Σ_j { log [(P_j + N_j) ^{(Pj + Nj)} /[P_j ^Pj N_j ^Nj ]  ] – C ε_j P_j })  .

The logarithm becomes

log [(P_j + N_j) ^{(Pj + Nj)} /[P_j ^Pj N_j ^Nj ]]

  = (P_j + N_j) log (P_j + N_j) - P_j logP_j  - N_j logN_j 

=  P_j [log (P_j + N_j) – logP_j] + N_j [log (P_j + N_j) – logN_j]

Then, the variation with respect to P_j of the above sum is

δ Σ_j {P_j [log (P_j + N_j) – logP_j] + N_j [log (P_j + N_j) – logN_j ] – C ε_j P_j }.

The variation of the sum is the sum of variations, so the variation symbol is brought inside the summation symbol, and the individual terms inside the summation become (using product and log rules of calculus)

(1)   δP_j log (P_j + N_j) +   P_j  δP_j /(P_j + N_j)  – δP_j logP_j  –  P_j δP_j / P_j 

(2)     0 + N_j δP_j /(P_j + N_j) – 0

(3)     – δ (C ε_j P_j) = – C ε_j δP_j

There are two terms in (1) cancel. The zeros in (2) are due to N_j not being a function of P_j, and in (3)  ε_j is not a function of P_j, so the variation doesn’t affect it. The summation becomes

Σ_j { log (P_j + N_j) +  P_j /(P_j + N_j) -  logP_j  - 1 + N_j /(P_j + N_j ) - C ε_j } δP_j = 0

=  Σ_j { log [(P_j + N_j)/ P_j] + ( P_j + N_j ) /(P_j + N_j) - 1 - C ε_j } δP_j

Or finally,

Σ_j { log [(P_j + N_j)/P_j ] - Cε_j } δP_j = 0.

Now, to paraphrase Bernard Lavenda, by using the variational process, we’ve made use of the constraint to “free” all the P_j‘s, so that the only way this equation can be satisfied is to set each coefficient of the δP_j’s equal to zero. (See Lavenda’s book Statistical Physics: A Probabilistic Approach, p. 22.)   That gives

 log [(P_j + N_j)/P_j ] - Cε_j = 0

Or

log [(1+ N_j /P_j ] = Cε_j.

The average energy (average amplitude-squared-energy of the oscillators at frequency ν_j) is by definition U_j = E_j /N_j , and we also have P_j = E_j  / ε_j ,  so the fraction N_j / P_j is equal to  ε_j / U_j , and

log (1 + ε_j / U_j )  =  C ε_j

1 + ε_j / U_j   =  exp[C ε_j]

U_j  =  ε_j / [ exp(C ε_j ) - 1 ].

With C set equal to 1/kT  and ε_j set equal to hν_j (see below), this is Planck's revolutionary average energy expression, found by using the calculations he didn't show in his December 1900 paper/presentation.  I, also, have not shown something I was planning to show in the above calculations—the entropy of an oscillator, or of all the oscillators at a particular frequency, or of ALL oscillators at all frequencies.  Without intending to, I bypassed entropy and went to average energy.  And I haven’t shown how to get C = 1/kT either, or illuminated the need for  ε_j being set equal to hν_j . I did do all that earlier, but I didn’t do it following Planck’s specific discussion in his December 1900 paper.  Maybe I’ll get around to it later!

The final expression for average oscillator energy is then

U_υ  =  hυ / [exp(hυ/kT) – 1],

and the long-sought spectral radiance expression is

σ (υ, T) = (8πυ²/c³) U_υ  =  (8πhυ³/c³) / [exp(hυ/kT) – 1]. 

Once he had a formula, Planck used it to calculate the values of the constants h and k numerically from the latest experimental data and the curves plotted from the data.  These were calculations he did in October, November, and early December 1900, and are likely what he later called the most strenuous work of his life. 

For the immediate consequences of Planck's work (as contrasted with the long term consequences), see Steven Weinberg's historical notes from his quantum class I took in 1998.  Also!  Remember that the discovery of the quantum was the beginning of what came to be called the wave-particle duality, and that quantum mechanics is not only about the particle-like discreteness of energy transformations, it is also about--mainly about--the wave-like interference of "probability amplitudes" involved in predicting the rates of emission and absorption of particle-like  packets of energy.  Planck stumbled upon what was later recognized as the first hint of the wave-particle duality in October 1900 when he first gave a derivation of his successful black-body spectrum formula, just before he discovered his quantum-energy-equals-constant-times-frequency formula. Nothing was done at the time with this hint of the wave-particle relation. A few years later Einstein brought it to light, using black-body radiation as his example.