Is Planck's Law Based on Incorrect Assumptions About Boltzmann's Distribution?

[Luck0]

So, my instructor said to us that Planck's law of radiation assumes that Boltzmann's distribution is incorrect. But it seems to me that Planck used Boltzmann's law, he just didn't replaced the summation by an integral, because now the energy is discrete. Can someone explain to me if my instructor is correct?

 

[Simon Bridge]

Your instructor is broadly correct.
Plank used the same initial argument that Boltzmann did.
What Boltzmann got wrong was to assume that energy could be infinitely divided: hence the difference.

 

[Jano L.]

So, my instructor said to us that Planck's law of radiation assumes that Boltzmann's distribution is incorrect. But it seems to me that Planck used Boltzmann's law, he just didn't replaced the summation by an integral, because now the energy is discrete. Can someone explain to me if my instructor is correct?

There are more derivations of Planck's formula.

In Planck's derivation the Boltzmann distribution is not used (he considers entropy of a set of material oscillators with quantized energy instead):

http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf

In Einstein's derivation, the Boltzmann distribution is used for the states of molecules:

http://astro1.panet.utoledo.edu/~ljc/einstein_ab.pdf

In Debye-Ehrenfest's derivation (the most common one found in textbooks) assumes validity of the Boltzmann distribution for discrete energies of the abstract EM oscillators.

I do not have original papers for this derivation, but check out the derivation of the formula 10.12 in section 10.7 of

http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf

There are other derivations, some of them may use Boltzmann's law, some of them may replace it by some other assumption on the connection of probability with temperature (like Planck did).

 

[dextercioby]

I tried to find articles by Ehrenfest or Debye which might support your claim, but couldn't. So how do you know they were the first ones to write it?

As a side note, the Boltzmann distribution is dated 1896 by M. Born in his <Atomic Physics> (2nd Ed., 1937), page 14 and used to derive Planck's radiaton formula on page 191. This means that this derivation had been produced by 1937.

In his book <The theory of Heat Radiation> (1914 Eng. translation of the 1913 German one), Planck sticks to the entropy-based derivation he put forward in the 1901 article in Annalen der Physik to whose English translation you have linked above.

In 1924 in his famous article, S. Bose mentions about <Since its publication in the year 1901 many types of derivations of this law have been suggested>. Later he goes to say that Einstein's famous 1917 article contains a <remarkably elegant derivation>.

The Einstein 1916/1917 famous article is to be found here: http://www.ulp.ethz.ch/education/quantenelektronik/Paper_Einstein2.pdf/

and indeed uses the Boltzmann distribution formula without resorting to the use of entropy.

So in a way, the <common> derivation known, I don't know if it can be attibuted to Ehrenfest and Debye, but definitely has its roots in the article of Einstein 1916/1917.

 

[dextercioby]

A full review article is on wiki:
http://en.wikipedia.org/wiki/Planck's_law#Planck.27s_views_just_before_the_empirical_facts_led_him_to_find_his_eventual_law

To be fair and square, Planck's original work, even with the use of entropy, started with Boltzmann's famous S = k ln W, so that somehow in each derivation you have classical statistics rooted there.

 

[dextercioby]

There's a comment in Andrade's book https://archive.org/details/structureoftheat001005mbp on page 356 about a certain derivation by Eddington without assumption of the Boltzmann's equation of exponential probability distribution:
A. S. EDDINGTON. On the Derivation of Planck s Law from Einstein s
Equation. Phil. Mag., 50, 803, 1925.

A useful read is here: http://master-mc.u-strasbg.fr/IMG/pdf/AJP_Lewis.pdf which includes in appendix C the mentioned derivation by Eddington, since Eddingtion's original article in not uploaded on the internet.

 

[Luck0]

Got it!

Thanks everyone!

 

[Jano L.]

I tried to find articles by Ehrenfest or Debye which might support your claim, but couldn't. So how do you know they were the first ones to write it?

As a side note, the Boltzmann distribution is dated 1896 by M. Born in his <Atomic Physics> (2nd Ed., 1937), page 14 and used to derive Planck's radiaton formula on page 191. This means that this derivation had been produced by 1937.

In his book <The theory of Heat Radiation> (1914 Eng. translation of the 1913 German one), Planck sticks to the entropy-based derivation he put forward in the 1901 article in Annalen der Physik to whose English translation you have linked above.

In 1924 in his famous article, S. Bose mentions about <Since its publication in the year 1901 many types of derivations of this law have been suggested>. Later he goes to say that Einstein's famous 1917 article contains a <remarkably elegant derivation>.

The Einstein 1916/1917 famous article is to be found here: http://www.ulp.ethz.ch/education/quantenelektronik/Paper_Einstein2.pdf/

and indeed uses the Boltzmann distribution formula without resorting to the use of entropy.

So in a way, the <common> derivation known, I don't know if it can be attibuted to Ehrenfest and Debye, but definitely has its roots in the article of Einstein 1916/1917.

Born's derivation in his book seems to refer to Planck's, but in contrast to Planck, Born uses Boltzmann distribution instead of entropy.

I read about Debye and Ehrenfest being the ones introducing the common derivation (quantization of EM modes) somewhere - probably some literature on history of physics or some older papers, I don't know for sure. These are probably the relevant references:

P. Debye, Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung, Ann, d. Phys. 33 (1910), 1427-1434.

P. Ehrenfest, Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?
Annalen der Physik 341: 91–118 (1911)

 

[dextercioby]

Thanks for the Debye reference, the original article is free to read on Gallica:

http://gallica.bnf.fr/ark:/12148/bpt6k153366/f1458.image.r=Annalen der Physik Leipzig.langEN

and the derivation of Planck's radiation law is also along Planck's lines, counting microstates starting with S= k lnW thus confirming what I wrote in post# 5, see page 1431.

Ehrenfest's article is also free to read on Gallica and on page 92 contains the famous phrase <Rayleigh-Jeans- Katastrophe I am Ultravioletten> :) On pages 95-96 he also uses Boltzmann's logic to count microstates with the purpose of computing the entropy, but his subsequent developments lead to a formula on page 108 which resembles the averaging procedure very much. So yes, based on this, we can somehow track this averaging procedure to this article by the Austrian P. Ehrenfest.

 

[dextercioby]

The exact derivation of the Planck formula using the Boltzmann distribution of energy levels and the summation of the geometric progression has been firstly given in 1911 by Planck himself, according to Max Jammer in his book <The conceptual development of Quantum Mechanics>, pages 52 and especially 53. Issue solved. :)