I would like to know how Planck went from an expression for the number of ways energy can be distributed in oscillators (denoted $W$) via the Boltzmann equation ($ S = k \ln W$) to the Planck radiation law - the distribution $B(\nu, T)$ of energy from a blackbody: $$ B(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{\exp(\frac{h\nu}{kT}) - 1}. $$ I know this is achieved by finding the number of states using combinatorics arguments to show that $W = (P+N-1)!/P!(N-1)!$, where $P$ is the number of energy quanta $\epsilon$ to be distributed over $N$ oscillators.

I think it hinges upon the relation between entropy and temperature: $$ \frac{dS}{dE} = \frac{1}{T}. $$

I am reading a history of quantum mechanics and want to see how quantising the oscillators emerged and how that then relates to later developments. As such, I specifically want an argument in the line of how Planck would have done it, rather than deriving the law from physics that didn’t exist in Planck’s time.

I’m okay with maths and technical details etc. - just really want to get my head around this.

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    $\begingroup$ Planck's sum is finite while the integral diverges; that's it. $\endgroup$ – sand1 Apr 29 at 9:20
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    $\begingroup$ I can't tell what you are asking. $\endgroup$ – Carl Witthoft Apr 29 at 14:00
  • $\begingroup$ I guess the question was a bit lengthy: I basically cannot see how to move from the factorial - which identifies how energy can be shared over oscillators - to the equation at the very top. But this is apparently how Planck did it. I believe this all hinges on using some statistical mechanics concepts but am too rusty to recall how you go from combinatorics to an expression involving temperature and a whole mess of constants. $\endgroup$ – a_former_scientist Apr 29 at 17:10
  • $\begingroup$ Can you access translations of Planck's paper? I can see them from Wikipedia. Planck, M. (1901). "Über das Gesetz der Energieverteilung im Normalspektrum". Annalen der Physik. 4 (3): 553. Bibcode:1901AnP...309..553P. doi:10.1002/andp.19013090310. Translated in Ando, K. "On the Law of Distribution of Energy in the Normal Spectrum" (PDF). Archived from the original (PDF) on 6 October 2011. Retrieved 13 October 2011. $\endgroup$ – M. Farooq Apr 29 at 23:20
  • $\begingroup$ Jammer M., The Conceptual Development of Quantum Mechanics, NY(1966; reedited'89) has nearly 10 pages of details about the derivation with comments and refs (he mentions where Planck differs from Boltzmann). The book is easy to find. $\endgroup$ – sand1 Apr 30 at 9:19

Thanks for all the comments. Using the paper linked by the helpful commenter (to which I found a pdf version, see here) - I could see how Planck did it (reminding me of all my old statistical mechanics in the process). Thought I'd sketch it out here to answer the OP.

We want to identify the energy distribution of a blackbody. Planck conceptualised the situation as a system of oscillators capable of absorbing and emitting quanta of radiation. In thermal equilibrium, he used some thermodynamic arguments to arrive at the distribution of energy of such a system.

First, define the overall energy of the system of $N$ oscillators as $E_N = NE$, where E is the average energy of one oscillator. Similarly the entropy is extensive: total entropy of the system is $S_N = NS$.

The quantisation step is to realise that to arrive at the known right answer (Planck found the energy distribution essentially empirically, before working out how to derive it) you must assume the energy stored by any oscillator to be quantised (not continuous) such that the total energy $E_N$ can be decomposed into $P$ units of a quanta of energy $\epsilon$: $E_N = P\epsilon$.

The number of ways of arranging the $P$ units across all $N$ oscillators is (as written above) given by the weight function: $$ W = \frac{(N+P-1)!}{P!(N-1)!} \simeq \frac{(N+P)!}{P!N!} \simeq \frac{(N+P)^{N+P}}{P^PN^N} $$ where we've assumed number of oscillators and number of energy units are large and then applied Stirling's approximation.

What I'd forgotten was the clever steps to utilise this within Boltzmann's equation (which was actually written by Planck for the first time) to get the entropy of the set of N resonators: \begin{align} S_N = k \ln W &= k \big[ (N+P)\ln (N+P) - N\ln N - P\ln P \big] \\ &= kN \big[ \left(1+\frac{P}{N}\right) \left( \ln N + \ln\left(1+\frac{P}{N}\right) \right) - \ln N - \frac{P}{N}\ln P \big] \\ &= kN \big[ \left(1+\frac{P}{N}\right) \ln\left(1+\frac{P}{N}\right) - \frac{P}{N}\ln \frac{P}{N} \big]. \end{align} Using $P = E_N/\epsilon = NE/\epsilon$ and dividing out the number of oscillators to get the average entropy per oscillator we get: $$ S = k \left[ \left(1+\frac{E}{\epsilon}\right) \ln\left(1+\frac{E}{\epsilon}\right) - \frac{E}{\epsilon}\ln \frac{E}{\epsilon} \right] $$

This is all framed in terms of the unknown energy quanta Planck introduced to be able to generate the form of the weight function he knew would lead to the correct result. To go beyond this he uses Wien's displacement law, which states that the peak blackbody intensity occurs at a frequency proportional to temperature of the blackbody - i.e. $\nu_{peak} \propto T$.

Planck goes through a series of steps (which I have to admit I haven't looked at in detail) to show that this implies that the entropy of a resonator sitting in a diathermic medium (which I think means basically in thermodynamic equilibrium) must be a function of $E/\nu$, i.e. $S = f(E/\nu)$.

Our expression for $S$ is framed in terms of $E/\epsilon$ which means that (up to a constant of proportionality) $\epsilon \propto \nu$, or $\epsilon = h\nu$ introducing Planck's constant. This 1901 paper (and a 1900 paper that seems to cover mostly the same ground) I think are some of the first places where the Planck constant is introduced and quantified - which I thought was quite cool!

Anyway, substituting this into the entropy equation we can find an expression for the energy of an average oscillator related to the temperature $T$: $$ \frac{1}{T} = \frac{dS}{dE} = \frac{k}{h\nu} \ln\left( \frac{1 + E/h\nu}{E/h\nu} \right) = \frac{k}{\epsilon} \ln\left( 1 + \frac{h\nu}{E} \right) \Rightarrow E = \frac{h\nu}{e^{h\nu/kT}-1}. $$ To go all the way to the energy radiance, it is possible to re-use the Rayleigh-Jeans law, namely that the radiance of a blackbody is $B_\nu = \frac{2\nu^2}{c^2}kT = \frac{2\nu^2}{c^2} E$, where $E$ is the average energy of a mode of radiation in the blackbody. This expression is incorrect since it assumes that each mode of radiation has (according to equipartition of energy) average energy $kT$. The reason I'm borrowing it here is because it contains the results of doing various integrals to deal with the 3D geometry so we don't have to (Planck refers to a previous paper of his to sidestep doing this in his paper).

As a result we can convert from energy of a mode of radiation to spectral energy radiance - i.e. the result quoted above in the OP - using Planck's expression for the average energy (as derived by considering the distribution of energy quanta across oscillators in the cavity wall), giving (finally) the result: $$ B_\nu = \frac{2\nu^2}{c^2} E = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}. $$ It is worth noting in the paper linked above that Planck writes out the spectral energy density not the energy radiance as I've done - the two can be switched between according to the relation: $u_\nu = \frac{4\pi}{c}B_\nu$.

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