Planck’s Proof of Planck’s Law vs New Proof


Max Planck took, as a leading proponent of the emerging German Empire at the turn to the 20th century, on the mission to solve the main open problem of classical electromechanics (interaction of matter and electromagnetic waves) of blackbody radiation, expressing his ambition as follows:

  • The spectral density of black body radiation … represents something absolute, and since the search for the absolutes has always appeared to me to be the highest form of research, I applied myself vigorously to its solution. 
  • The German people ha found itself again. One thing only we know, that we members of our university…will stand together as one man and hold fast until – despite the slander of our enemies – the entire world comes to recognize the truth and German honor. (1914)

Planck eventually came up with a solution named Planck’s Law after what he has described as the hardest struggle of his life:

  • …the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be… 
  • My maxim i always this: consider every step carefully in advance, but then, if you believe you can take the responsibility for it, let nothing stop you. 
  • My futile attempts to fit the elementary quantum of action into classical theory continued for a number of years and cost me a great deal of effort. Many of my collegues saw in this something bordering on a tragedy.
  • Hitherto the principle of causality was universally accepted as an indispensable postulate of scientific research, but now we are told by some physicists that it must be thrown overboard. The fact that such an extraordinary opinion should be expressed in responsible scientific quarters is widely taken to be significant of the all-round unreliability of human knowledge. This indeed is a very serious situation. 
  • Despite the great success that the atomic theory has so far enyoyed, utimately it will have to be abandoned in favor of the assumption of continuous matter (wave mechanics) (1882). 

In the previous post Two Proofs of Planck’s Law vs Backradiation I compared Planck’s proof of Planck’s Law with the new proof I have presented in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation.

Planck’s proof is based on statistics of quanta of energy similar to that used in statistical mechanics, while my new proof is based on classical deterministic wave mechanics combined with a certain form of finite precision computation.

Planck’s worked out different versions of his proof with the most readable in Lectures 5 and 6 of Eight Lectures on Theoretical Physics (1909) followed by the lengthier The Theory of Heat Radiation (1914). Let us first take a look at Planck’s Proof as presented in Lectures 5 and 6 and compare with my New Proof (with a comment on The Theory of Heat Radiation below).

Both proofs start from a model of a blackbody as a vibrating string (system of resonators) subject to small radiative damping interacting with an exteriors forcing:

  • U_{tt} - U_{xx} - \gamma U_{ttt} = f ,

where the subindices indicate differentiation with respect to space x and time t, and

  1. U_{tt} - U_{xx}: material force from vibrating string with U displacement
  2. - \gamma U_{ttt}Abraham-Lorentz (radiation reaction) force
  3. f: exterior forcing.

The basic result of both proofs is the following energy balance in stationary state, established by a spectral analysis assuming periodicity in space and time:

  • \int \gamma U_{tt}^2dxdt = \epsilon \int f^2 dxdt    (Rayleigh-Jeans Law)

stating that the radiated energy is equal to the energy of the forcing, where \epsilon\approx 1 is a coefficient of absorptivity/emissivity. This is a non-trivial result which can be viewed as Rayleigh-Jeans Law, and which is more carefully exposed in New Proof than in Planck’s Proof, but the essence is the same, and the essence is:

  • near-resonance in a resonating system with small damping.

What comes, goes in and then goes out: simple but profound.

The trouble with Rayleigh-Jeans Law is that it does not show the high-frequency cut-off seen in measurements: Sufficiently high frequencies of the forcing are not re-radiated (but instead contribute to internal heating of the resonators as an important feature of New Proof missing in Planck’s Proof only focussing on emission).

The proofs use different mechanisms to effectively introduce the required high-frequency cut-off:

  1. Planck’s Proof uses statistics of energy quanta.
  2. New Proof uses deterministic dissipation as a model of finite precision computation.

New Proof thus stays close to the original deterministic wave model by introducing a small dissipative effect which can be motivated on physical grounds in the same way as inviscid flow is “regularized” by adding small viscosity. Internal heating is the result of the dissipation.

Planck’s Proof leaves the realm of deterministic wave mechanics and introduces a seemingly ad hoc model of statistics with the effect of internal heating being obscure. It is easy to understand that this step filled Planck with horror, since his soul was trained with classical deterministic wave mechanics as a basis physics. To resort to statistics was a failure and the cost has shown to be very high: A wave model with small damping/dissipation is understandable because it mimics understandable physics, while a statistical model thrives from the fact that the physics is not understandable.

In short: New Proof is (easily) understandable, while Planck’s Proof is very difficult to understand.

The sad result is that physicists of today refuse to talk about Planck’s Proof, and thus refuse to have any opinion on the physics of “backradiation” carrying CO2 alarmism supposed being connect to Planck’s Law.

This is troublesome because it leaves room for non-physicists to tell truths about physics which are not true. And they do.

Planck would most likely have welcomed New Proof, as a relief from the agony of statistics. Physicists of today don’t suffer like Planck did, because they have been trained to love statistics and frown at deterministic wave mechanics.

The lesson learned in this exercise is that in order to understand a mathematical result (like Planck’s Law), it is necessary to understand the mathematical proof of the result. The result is a summary of the proof and the proof is the real story. It is impossible to properly understand The Idiot by Dostoyevsky by reading a short summary on a couple of lines.

Planck’s Law is fundamental in climate science, but there are different views concerning the proper meaning of Planck’s Law, specifically whether it supports a phenomenon of “backradiation” or not. The quarrel can only be settled by carefully scrutinizing a proof of Planck’s Law. I have tried to do so, and my conclusion is that Planck’s Law does not support any idea of “backradiation”. I would appreciate if someone else would show interest in the proof of Planck’s Law.

Comment on The Theory of Heat Radiation

Planck starts out in the Preface with:

  • the theory thus developed does not by any means claim to be perfect or complete…

and then sets the foundation on page 1:

  • All heat rays which at a given instant pass through the same point of the medium are perfectly independent of one another.

This statement can be read as a support of “backradiation” with two-directed “heat rays” between two blackbodies. But was Planck in 1914 correct in claiming the existence of many-directed “heat rays” as the physics of radiation?

Well, to someone believing that busy little photon particles carry heat back an forth, it is the perfect support.

To someone like me believing that instead IR is waves rather than particles, Planck’s statement represents a trivialization or infantilization of physics: To speak about “heat rays” as the basis of radiative heat transfer is black magic and not physics.

Planck thus starts out in a wrong direction and the rest of his treatise is a lengthy discussion of with the usual elements of statistical mechanics of entropy, probabilities based on the idea of energy quanta.

Gone is the the wave equation of Lectures 5 and 6 from 1909, now replaced by “heat rays” of quanta. Planck’s resolution of the “ultraviolet catastrophe” of classical wave mechanics has turned into a catastrophe for physics.

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