# Talk:Planck's law/Archive 8

## Maxwell-Boltzmann distribution does not occur in nature but is only an illusion in the minds of bears of little brain

The Maxwell-Boltzmann distribution does not describe anything that actually occurs in nature. It is an illusion in the minds of bears of little brain that arises because they imagine that the low density limit can actually be achieved in nature. They foolishly imagine that with the actually occurring achievement of the limit, molecules in a gas in thermodynamic equilibrium obey the Maxwell-Boltzmann distribution, and behave as particles that move freely between collisions. Bears of little brain make observations of tracks in cloud chambers, observing what seem to them like particles in the actually occurring low density limit which they foolishly and ignorantly analyze, often using Maxwell's laws to identify them, as if they were tracks of freely moving classical particles.

But those who are expert in quantum theory, smarter than the average bear, know that there are really no free particles because the low density limit is ony a theoretical figment, and does not occur in nature. So they know that the Maxwell-Boltzmann distribution is only a theoretical figment not occurring in nature. This enables them to know that the S matrix formalism of quantum field theory is physically sound because black is white, a secret understood only by experts in quantum theory, but baffling to bears of little brain.

That ignorant fool Niels Bohr mistakenly thought that quantum mechanics was based on experimental observations. Even worse, he mistakenly thought that it is impossible to know exactly what happens inside a system of bound particles or inside a quantum reaction region, but that for the physics of such things as molecules, it is possible to make observations only by use of laboratory experimental apparatus, which is macroscopic. His foolish idea was that a quantum mechanical experiment is done by starting with a macroscopic source of a beam (e.g. a lead box containing some radium, or a remote star seen through a telescope). A hole is made in the side of the box, so small that the beam issuing from it will not saturate the particle counters that are to be used for the experiment, but large enough to supply a beam that is stronger than the noise levels of the counters. If the hole were too big, so as to let the beam saturate the counters, then the quantum counting would not be possible. The beam is then passed in succession to an initial quantum analyzer, such as a Stern-Gerlach apparatus, to an array of reaction regions (one for each output channel of the initial quantum analyzer) such as electric fields, to an array of final quantum analyzers, and at last to the particle counters. Foolish and ignorant Bohr thought that he did not know what went on inside the macroscopic source box nor in the reaction regions, but that his only knowledge of the quantum experiment arose from his knowledge of the macroscopic features of the reaction region and from the counts of the particle counters, which are macroscopic laboratory devices.

The quantum analyzers are essential to this foolish dream of Bohr. They are also macroscopic laboratory devices. They work on the same general principle as the Stern-Gerlach apparatus. A particle in the entering beam enters into a weak and unstable quantally bound state with the Stern-Gerlach magnet (as if a giant molecule in a nearly classical state of excitation) and leaves again in a definite quantized state, one of several possible discrete states defined by the macroscopic setting of the magnet. The general state of the particle in the beam is hardly affected by such very weak binding, except for the quality by which the analyzer splits the beam into discrete exit sub-beams. That ignorant fool Heisenberg agreed with the ignorant fool Bohr about most of this. Those ignoramusus thought that the spin of the fermion was like a clock with no twelve o'clock marked on its face, so that you could never know the exact phase of a quantum harmonic oscillator. Accordingly they introduced complex numbers into the analysis, and so they constructed the density matrix, including off-diagonal elements, from the statistics from the macroscopic experimental particle counters. From the density matrix and the idea of the quantum harmonic oscillator they could by strict logic derive ideas about the wave function as explanatory of the experimental observations. In contrast, those smarter than the average bear know that the primary reality is the wave function obeying the Schrödinger equation, and that Niels Bohr's bizarre idea, that one learns about physics by doing experiments with macroscopic apparatus, is just a dream.

Discrete quantum variables are definable as detectable by quantum analyzers of this general kind. If such an analyzer can be found for a physical characteristic, then experiment justifies the view that it is discretely quantized.

There are also analyzers, such as prisms for light, that analyze into continuum spectra. Bears of little brain think that light as analyzed by a prism has actually achieved the low density limit and that photons move as free particles. They are even foolish enough to think that photons are the behaviour of the oscillating electromagnetic field in the far field, though of course that is only a figment of their silly imaginations, because those smarter than the average bear know that no field is far enough to actually reach the limit.

Such a discrete quantum analyzer is known to exist for cats, so that a beam of cats can be passed through it. By this analyzer, the beam of cats is split into exactly two sub-beams, dead and alive. The state of health of the cats is hardly affected by the analyzer, except for the decision as to its being alive or dead. This has been checked by the animal experiments ethics committee. These cat live/dead analyzers are two-a-penny in the laboratories of those smarter than the average bear. That is why it is well known that for cats the dead/alive distinction is a two-valued discrete quantum variable.

Moreover, as a result of experiments performed by those smarter than the average bear, it is known that the entire universe can be put through such a quantum analyzer. In fact this has often been done in their laboratories. They have quantum analyzers larger than the entire universe, through which they pass beams of entire universes in order to check their discrete quantum states. The settings on such entire-universe analyzers are set by those smarter than the average bear (who are very big chaps), and the knowledge of the results gives them omniscience and infallibility, so that they can with omniscience and infallibility say what is correct and what is not. We are fortunate to have some of these big chaps in our editorial team here. These are the chaps who know that the Maxwell-Boltzmann distribution does not occur in nature because there are no freely moving particles in nature, unless, in the laboratory next door to the quantum mechanics laboratory, such freely moving particles are used for quantum field theory experiments, both as initial physical input to the reaction region and as output from it, into the laboratory particle counters. Like that ignorant fool Niels Bohr, the quantum field theorists say that they do not know exactly what happens in the interaction region, because they regard observation inside that region as precluded. Nevertheless, the quantum field theorists also are omniscient and infallible, for reasons like those given above.Chjoaygame (talk) 22:31, 10 May 2012 (UTC)

This fool prefers much more straightforward language. What are you trying to say? Keep it short please. Thanks. Dmcq (talk) 22:44, 10 May 2012 (UTC)
I am saying that I think you are not a fool.Chjoaygame (talk) 22:50, 10 May 2012 (UTC)

Chjoaygame edited this paragraph in the article:

"The Planck distribution is the unique stable distribution for electromagnetic radiation in thermodynamic equilibrium.[4] It takes its structure from the Bose–Einstein distribution, a fundamental energy distribution for immaterial 'particles' such as photons and phonons and gluons, which are not conserved, in thermodynamic equilibrium. There are two other fundamental equilibrium energy distributions: the Fermi–Dirac distribution, for quantally bound elementary particles, such as electrons, and the Maxwell–Boltzmann distribution for freely moving unbound material molecules."

I rewrote this as follows:

"The Planck distribution is the unique stable distribution for electromagnetic radiation in thermodynamic equilibrium.[4] It takes its structure from the Bose–Einstein distribution, a fundamental energy distribution for bosons in thermodynamic equilibrium. In case of immaterial 'particles' such as photons and phonons and gluons, which are not conserved, the chemical potential is zero, which reduces the Bose-Einstein distribution to the Planck distribution. There is another fundamental equilibrium energy distributions: the Fermi–Dirac distribution, which describes fermions, such as electrons, in thermal equilibrium. The difference between the two distributions derives from the fact that while multiple bosons can occupy the same quantum state, fermions cannot. At low densities where the number of available quantum states is much larger than the particle density, this difference becomes irrelevant. In the low density limit, both the Bose-Einstein and the Fermi-Dirac distribution reduce to the Maxwell–Boltzmann distribution."

Then Chjoaygame edited that paragraph again, correcting a typo made by me but also putting back the incomprehensible sentences about "freely moving unbound material molecules:

"The Planck distribution is the unique stable distribution for electromagnetic radiation in thermodynamic equilibrium.[4] It takes its structure from the Bose–Einstein distribution, a fundamental energy distribution for bosons in thermodynamic equilibrium. In case of immaterial 'particles' such as photons and phonons and gluons, which are not conserved, the chemical potential is zero, which reduces the Bose-Einstein distribution to the Planck distribution. There is another fundamental equilibrium energy distribution: the Fermi–Dirac distribution, which describes fermions, such as electrons, in thermal equilibrium. The two distributions differ because multiple bosons can occupy the same quantum state, while multiple fermions cannot. At low densities, the number of available quantum states is much larger than the particle density. In effect the molecules are unbound and freely moving, and this difference becomes irrelevant. In the low density limit, both the Bose-Einstein and the Fermi-Dirac distribution reduce to the Maxwell–Boltzmann distribution, which describes the effectively free motion of molecules in a material gas."

And then I removed these sentences again:

"The Planck distribution is the unique stable distribution for electromagnetic radiation in thermodynamic equilibrium.[4] It takes its structure from the Bose–Einstein distribution, a fundamental energy distribution for bosons in thermodynamic equilibrium. In case of immaterial 'particles' such as photons and phonons and gluons, which are not conserved, the chemical potential is zero, which reduces the Bose-Einstein distribution to the Planck distribution. There is another fundamental equilibrium energy distribution: the Fermi–Dirac distribution, which describes fermions, such as electrons, in thermal equilibrium. The two distributions differ because multiple bosons can occupy the same quantum state, while multiple fermions cannot. At low densities, the number of available quantum states is much larger than the particle density, and this difference becomes irrelevant. In the low density limit, both the Bose-Einstein and the Fermi-Dirac distribution reduce to the Maxwell–Boltzmann distribution."

I rest my case. Count Iblis (talk) 23:07, 10 May 2012 (UTC)

What in the world is that rambling all about? How is it even remotely controversial to explain the various links between the MB/BE/FD and Plank distributions, and that the links only exist in the minds of the foolish? 01:00, 11 May 2012 (UTC)

While I fully realize saying this is not in the spirit of collegial collaboration between editors, I bet this article would gradually improve over time if both Chjoaygame and Headbomb were totally banned from it. I've never seen either of them make an edit that could be considered a net benefit to this article. It's like Asperger's meets Mafia! --Vaughan Pratt (talk) 08:00, 11 May 2012 (UTC)

To expand on this a little, Headbomb as usual has no idea what Chjoaygame is saying, but that's not unique to Chjoaygame, so far I've not been able to identify anyone HB is particularly sympatico with. Chjoaygame on the other hand has a very in-depth appreciation for the underpinnings of Planck's law, but one whose logic seems in recent months to be falling apart. In his monologue above he says "That is why it is well known that for cats the dead/alive distinction is a two-valued discrete quantum variable." However this view has of late been falling into disrepute for humans (and hence presumably also for cats) in view of problems with the single-minded emphasis on the brain as the only reliable criterion for life, see e.g. http://www.cmaj.ca/content/164/6/833.full. Chjoaygame's logic could benefit from responding faster to changing premises, right now it seems to be at risk of getting stuck in the mud. --Vaughan Pratt (talk) 08:58, 11 May 2012 (UTC)

• I've trimmed and re-written those paragraphs, see what you think. Waleswatcher (talk) 13:50, 11 May 2012 (UTC)
• I have reported Vaugan Pratt to the Ministry of Truth for the nearly unspeakable crime of scepticism. He is expressing doubt about what I have indicated has been checked as correct by the animal experiments ethics committee, and he even seems to doubt that those smarter than the average bear really have cat live/dead quantum analyzers at two-a-penny in their laboratories. Shocking!Chjoaygame (talk) 23:15, 11 May 2012 (UTC)
The characterization of the geometry of the Stern-Gerlach regime as binary might indeed be called the Winnie-the-Pooh characterization---if that name ever catches on Chjoaygame can take credit for it. Later on one learns that this regime is a good deal more nuanced, see e.g. Schleich, Quantum Optics in Phase Space, Chapter 20, "Wigner Functions in Atom Optics." The subtle nature of the regime is visible in the Stern-Gerlach experiment itself when the lateral component is taken into account, where the pattern is not a simple pair of dots but rather a broad-lipped smile. --Vaughan Pratt (talk) 23:17, 12 May 2012 (UTC)

## The start of the article is pitched at far too high a technical level

The start of the article is pitched at far too high a technical level. In particular those readers interested in what the law is, what it means, and how to apply it, will get very little from the "Physical outline" section about that, and to the extent these appear anywhere else in the article they are hard to find and collect together. Does anyone have any objection to inserting a more elementary section ahead of "Physical outline," labeled "Introduction" or whatever people think is appropriate, containing the material in the first section at User_talk:Vaughan_Pratt/Planck's Law? Normally I'd be bold and just insert it without asking, but this particular article went through so many edit wars in recent months that it seems advisable under the circumstances to ask first.

I also have some questions about this physical outline section, but I'd like to address the more basic issues first. --Vaughan Pratt (talk) 20:06, 11 May 2012 (UTC)

I agree, that section could be moved further down in the article. Count Iblis (talk) 20:23, 11 May 2012 (UTC)
The physical outline section originally was written at an introductory level suitable for bears of little brain, and was intended to introduce the main physical ideas ahead of the detailed presentation of the formulas.
It has now been lifted by Count Iblis and Waleswatcher to the level of those smarter than the average bear, and as a direct result some editors agree that it should no longer serve as an introduction for bears of little brain, because it is too technical.
It seems I was mistaken to mention the Fermi-Dirac and Maxwell-Boltzmann distributions at that introductory stage. They had been in the lead but nowhere else in the article, and I just lifted them to the physical outline section; a mistake, it now emerges. Not the main focus of Planck's law, they are commentary of special interest to those smarter than the average bear, whose enthusiasm originally put them directly into the lead with no other mention of them in the article.
The physics of bosons and fermions and the limiting case of many available states could well be presented in the physics section further down, as Count Iblis suggests, but they are not introductory material for bears of little brain.
Vaughan Pratt wants to start by taking the widest possible range of applications of the law (cosmic microwave background, black bodies out of thermodynamic equilibrium, etc.). This would obscure the physics of the law. Vaughan Pratt sees the law primarily as a formula. I see it primarily as physics. Vaughan Pratt is bursting to tell us how he analyzes the formula as a formula (how the maximum depends on the choice of spectral variable, etc.), forgetting the physics. That is part of why I talk about trojan horses. To support his plan, Vaughan's red font enthusiasm on his talk page in attacking my physical statements is remarkable.
The physics is very important. Some introductory outline of the physics, more or less suitable for bears of little brain, should be given before the mathematical details of the formulas.Chjoaygame (talk) 23:03, 11 May 2012 (UTC)
I have several comments on your second last paragraph concerning my intentions. First, you want to start by stating the law for a very restricted situation. This is like introducing the counting concept as being for the purposes of counting sheep, and complaining about removing that restriction with "you want to start by taking the widest possible range of applications of the counting concept." All I'm proposing is to remove your restriction. That doing so makes the law applicable to a much wider range of situations is a good thing, not a bad thing. It would only be bad if one allowed an enumeration of examples of such situations to overburden the article with TMI.
Second, details such as cavities and properties thereof (solid walls) are a red herring, making them unnecessary for the very first sentence of the article. So is thermodynamic equilibrium: the law is equally valid for a radiator radiating freely into space with no returning radiation and hence no equilibrium. Granted the radiation will decrease as the radiator thereby cools, but at no instant during the cooling does it violate Planck's law.
Third, you've somehow formed the impression I'm a formalist when it should be obvious from my writing that I'm a Platonist. While there are lots of formulas for Planck's law, and 2-3 graphs for it, there is only one law, namely a function of temperature and position in the spectrum. Its value is the energy radiated per unit area of radiator, equivalently the power radiated per unit area of radiator per unit frequency bandwidth of spectrum. (Solid angle is a distraction because black bodies are Lambertian whence there is no essential difference between exitance and radiance relevant to the law other than a constant factor of π in Planck's law. The article should make this clear.) Likewise there is only one Wien approximation and one Rayleigh-Jeans law, and Planck's law is their reconciliation, and a remarkably simple and elegant one given how different the two laws look at first!
Regarding your I see it primarily as physics, where we differ is in the scope of "physics." One might want to argue that when Planck first found his law he was not doing physics, but that would be a philosophical sort of argument. Whatever he was doing, I think we could both agree he wasn't doing rheology or metallurgy. What you're calling the "physics" of Planck's law includes such components as its idealized phenomenology and its statistical mechanical derivation. Whether these should go in the first sentence is a good topic for discussion at this page; I interpret CountIblis as voting against, as do I. No one else has weighed in with a preference so far, whence my reluctance to edit this article given its recent history of vigorous edit warring.
Wish I knew what to do about your "trojan horse" and "chatty" objections, sorry I'm unable to help there. --Vaughan Pratt (talk) 02:42, 13 May 2012 (UTC)
When a restriction is removed from the restricted statement of the law, it should be clearly and explicitly shown how and why the removal of the restriction does not lead to departure from the law, or if it does lead to departure from the law, some indication should be given of how and why and by how much.
Real radiation from a cooling body comes from various depths from the surface of the body and they will in general have various temperatures, especially if the surface is very rough and the material is a poor conductor of heat. This should be taken into account as one of the possible reasons for small departures from ideality of experimental tests of the Kirchhoff law, and must cause some departures from ideal conformity to the Planck law, even if very small. (references Salisbury, J.W. Wald, A., D'Aria, D.M. (1994), Thermal–infrared remote sensing and Kirchhoff's law. 1. Laboratory measurements, J. Geophys. Res. 99, 11897–11911; Hapke, B. (1993), Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, ISBN 0–521–30789–9.) The effect may be nearly negligible for many practical purposes, but is detectable in the laboratory and some account of it should be taken for an article on matters of physical principle.
I disagree with your statement that the physical details are a red herring. They often contain some of the physics, which I regard as important.
I haven't classified or labeled your philosophical viewpoint. I don't know what a "formalist" is.Chjoaygame (talk) 08:17, 13 May 2012 (UTC)

## response of VP

When a restriction is removed You misunderstood me, I wasn't proposing to remove the restriction "ideal black body," I was only proposing to remove unnecessary details about their definition or nature, which are more appropriately dealt with in the black body article, and/or further down in this article. My proposed introduction removes no restrictions and is quite clear that the law is restricted to ideal radiators.

Real radiation from a cooling body comes from various depths from the surface of the body Yes, exactly. Yet when I gave limb darkening as an example of this phenomenon in my proposed introduction you argued against its inclusion. What is your objection to pointing out ways in which nominally black bodies like the Sun can fail to be perfectly black?

the physical details ... often contain some of the physics, which I regard as important. Yes, so much so in fact that you are "bursting," as you put it, to get them into the first sentence. There's a place for everything that's important, but it's unreasonable to expect that everything that's important has to go in the first sentence!

I don't know what a "formalist" is. A formalist is someone who sees the law as a formula. A Platonist is someone who sees it as a function. You wrote Vaughan Pratt sees the law primarily as a formula, thereby labeling me as a formalist. Happy to explain the difference if it's unfamiliar.

Many people depend heavily on Planck's law in various practical situations, for example global warming (see the first third figure at greenhouse effect). When they come to this article to learn about Planck's law it is a distraction for them to learn that thermodynamic equilibrium is a prerequisite, particularly if they've just been reading that global warming is a disequilibrium phenomenon. --Vaughan Pratt (talk) 20:29, 14 May 2012 (UTC)

### response

I started intending to respond in detail, and indeed wrote a detailed response. But now I think it would not have been wise for me to post it. Chjoaygame (talk) 23:23, 14 May 2012 (UTC)

## plans

I've duplicated Chjoaygame's proposed rewrite at User talk:Vaughan Pratt/Planck's Law so that I could annotate it without breaking up the flow of the original. This includes some remarks aimed at clarifying the need for things like "Roughly speaking" in my version. (I calculated the error from applying Planck's law directly to a cone of solid angle π without applying Lambert's law, using the formula 2π(1 − cos(θ)) for the surface area of a spherical cap of angular radius θ, which turns out to be θ = 60° for one steradian, and got a factor of 3 which surprised me, I was expecting just a bit more than 1! Someone please confirm or disconfirm. Perhaps Chjoaygame was complaining that I should have said "Very roughly speaking.") There is also an error from using 1 Hz instead of but it's tiny by comparison with the error from using 1 steradian in place of dΩ! Any error from using unit area instead of dA exists only when A is nonplanar; only a mathematician could imagine a dA that was not planar. :)

There's also a section discussing my graph, which User:Q Science felt could be improved on but it turns out not to be that simple. That's not needed for now anyway since the current goal is just to replace the table with a reasonably short section (four items in my current proposal, namely statement, explanation, Lambert's law, and the wavelength form) that has incorporated everyone's suggestions for improvement. Otherwise that section will remain in a permanent state of flux while people argue about what it should look like. --Vaughan Pratt (talk) 21:19, 10 May 2012 (UTC)

• I would prefer that people not calculate anything new please if possible. An example is the only place I'd wish a persons own calculations. Surely there are enough books about it to be able to summarize something out of. Putting in one's own formulae is the sort of thing I associate with someone who seeming some people here don't want to mention. Dmcq (talk) 22:49, 10 May 2012 (UTC)
Those wanting to see the current table go away are 100% in agreement with you! In order to create a notational conflict with Boltzmann's constant k in the article back in October that would prevent people from reverting his kB back to k, Headbomb originally thought it would suffice to just directly replace ν̃ with k. However after the difference between wavenumber and angular wavenumber had been explained to him, and then after some false starts, he was finally able to calculate the correct formula for the latter. (Had he known the Stefan-Boltzmann fourth-power rule for scale conversion in Planck's law he wouldn't have found it so hard.) But that formula too was on the verge of being deleted until it occurred to him to build an elaborate table that would justify inclusion of the angular wavenumber formula and hence of its associated variable k so as to maintain the notational conflict. I'd have taken him to ANI over this if I'd had the time back then -- this sort of gaming the system is all very time consuming for editors who simply want to get on with the job. --Vaughan Pratt (talk) 07:50, 11 May 2012 (UTC)
• Here is a copy-and-paste from Vaughan Pratt's user talk page:
The following three sections are for future discussion, including the figure (apropos of which see User:Q Science's comments on it and my responses near the bottom of this page). If the preceding section is not adopted the following sections become moot as they were written under the assumption that the above material has been presented.
As I read this it perhaps implies that, as things are written at present, if the "preceding section" is agreed to then such agreement might be interpreted as some kind of consent to consider the "following three sections". That's one aspect of my trojan horse feeling.Chjoaygame (talk) 04:27, 14 May 2012 (UTC)
You're interpreting the conditional in the wrong direction. Writing A for inclusion of the introduction and B for inclusion of the rest, what you quoted from my talk page was not-A → not-B (equivalently B → A). You're reading it as the converse, A → B, equivalently not-B → not-A. Had the latter been the meaning it would make sense to block A as a way of avoiding B, which is what you seem to be arguing for, but it wasn't. --Vaughan Pratt (talk) 16:43, 16 May 2012 (UTC)

## The edit I reverted

said: "Photons can travel at only one speed, that of light, so that the differences between them are described by just one factor, the energy or its equivalent, the frequency...For a material gas at given temperature, the pressure and energy density can vary independently for different gases, because molecules can travel at a range of speeds"

This is incorrect, or at least very misleading. For either a photon or a particle of a given mass, the speed is uniquely determined by the energy or momentum, and only one number (the energy or momentum) is needed to describe the relevant differences between different particles. Perhaps what's meant is that different gases have constituents with different masses, but that's not what's said. Waleswatcher (talk) 23:33, 11 May 2012 (UTC)

Yes, that's what's meant.Chjoaygame (talk) 01:07, 12 May 2012 (UTC)
I didn't understand this reversion, I thought Chjoaygame was exactly correct here (note that he was only talking about a photon gas, the comparison with other gases comes later and you didn't revert that). Assuming the photon is traveling in a vacuum, its speed is not determined by anything, not even the frame of reference, because it is always c. The photon's energy determines only its frequency, not its speed. Its momentum as the determinant of pressure is the quotient of its energy by c. (That the denominator is so large is why it took half a century to measure the pressure after it occurred to Crookes, Maxwell, and Reynolds that light could exert pressure, which however they had no way of estimating in the 19th century. Failure to measure the pressure of light back then was like Newton's failure to measure its wavelength a century and a half earlier: both were too small to observe with the respective understanding and technologies of the day. One difference is that the smaller the wavelength the larger the pressure.)
It's not that I object to this stuff, btw, I only object to having the first section devoted to it. It's all beautiful stuff, but as I said it is at way too high a level for the introductory section. If people agree with my proposal to insert my suggested six-paragraph introduction ahead of this I'll do so, but only if, since this article has had more than its share of edit wars recently.
An alternative would be to regard this as the main article on Planck's law, conducted at a suitably high level, and to have a much shorter introductory article at a level that could be readily followed by those with only high school physics. Wikipedia has lots of introductory articles supplementing main articles in this way. If there's a sense that this would be preferable I'd be happy to write such an article. Either way is equally fine by me. --Vaughan Pratt (talk) 03:19, 13 May 2012 (UTC)
Of course photons move at c. What was incorrect was the inference drawn from that. See my previous comment. Waleswatcher (talk) 10:49, 13 May 2012 (UTC)
What you wrote was "Perhaps what's meant is that different gases have constituents with different masses, but that's not what's said." But it was said, in the comparison with "material gases, for which the pressure and energy density depend on the molecular masses and other characteristics." Absent that dependence you're right, but t That's why Chjoaygame put in the dependence (mass presumably being the main characteristic, not sure what other significant ones there are). --Vaughan Pratt (talk) 18:26, 13 May 2012 (UTC)
I overstruck "Absent that dependence you're right" just now because not only was Chjoaygame's fix correct, but there are problems both with what it was fixing (which are still there now) and with your "For either a photon or a particle of a given mass, the speed is uniquely determined by the energy or momentum..." There is a crucial difference between photons and molecules, namely that the energy does not determine the speed of the former, only of the latter; furthermore whereas the momentum of the former is h (= ħ), for the latter it is mv where m is the rest mass of the molecule. Moreover what Chjoaygame was (correctly) trying to fix was "Because all photons are identical, the pressure and energy density of a photon gas at equilibrium are entirely determined by the temperature." While the consequent is a correct statistical mechanics statement, the antecedent is not: photons are identical only in speed, not energy. Molecules of a homogeneous gas are identical only in mass, not speed. Perhaps there is a better fix for this than the one Chjoaygame came up with though I don't see it right now. --Vaughan Pratt (talk) 19:49, 13 May 2012 (UTC)
Nope. Once you know the mass, the energy or momentum determines the speed. That holds both for the case of zero mass and for the case of non-zero mass. There is nothing at all special about photons in that regard. What I think was meant was that you might not know the mass, or you might have a mixture of constituents with multiple masses, or something. But what was written wasn't that, and it was wrong. Waleswatcher (talk) 00:28, 14 May 2012 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Yes, the relevant difference is simply that the chemical potential is zero for the photon gas. This means that you have one less thermodynamic variable compared to a gas of molecues, where the total number of particles can be chosen freely. Count Iblis (talk) 00:44, 14 May 2012 (UTC)

What I think was meant was that you might not know the mass, or you might have a mixture of constituents with multiple masses, or something. But what was written wasn't that, and it was wrong. What was written was "This is unlike the case for material gases, for which the pressure and energy density depend on the molecular masses and other characteristics of the constituent particles." How is that different from "you might not know the mass?"
Also you seem to be using "determine" differently from me. If B is independent of A I wouldn't say A determines B, but apparently you're ok with that in the case when B is the speed of a photon and A is either its energy or momentum.
But perhaps Chjoaygame should be the judge of whether his reverted edit is more accurate than what it was reverted back to. Seems to me it is but if he thinks otherwise I won't press the point. --Vaughan Pratt (talk) 06:21, 14 May 2012 (UTC)
I am no judge. It seems to me that when Waleswatcher says "all photons are identical", he means that they are of only one kind, but can exist in different states of excitation, frequency, or energy?Chjoaygame (talk) 09:13, 14 May 2012 (UTC)
Quite right, that's what he means. But if that were the standard meaning then "two identical photons" would mean the same as "two photons." If you Google for "identical photons" you'll see it's a widely used concept that never means that! This article is using a nonstandard meaning of "identical" for photons, and moreover without saying so. You were right to remove "because all photons are identical." Waleswatcher's complaint was that in removing it you didn't explain that "different gases have constituents with different masses." I could have sworn your wording was equivalent, but if not then the correct fix would have been to replace your wording with Waleswatcher's, not to revert back to the problematic "all photons are identical." --Vaughan Pratt (talk) 17:15, 14 May 2012 (UTC)
I don't know why this is so confusing. What was written was "For a material gas at given temperature, the pressure and energy density can vary independently for different gases, because molecules can travel at a range of speeds." That was being contrasted to the case of a photon gas, where "Photons can travel at only one speed, that of light, so that the differences between them are described by just one factor, the energy or its equivalent, the frequency". For a material gas composed of a given type of molecule with some definite, fixed mass (and various excitation levels if we are discussing high temperatures), the pressure and energy density do not vary independently, and the differences between individual molecules in the gas are described only by one factor, the energy or momentum - just like for photons. Photon gases are simply the specific case of that where m=0. It's true that when comparing different gases composed of molecules or particles with different masses, then the mass enters into the various thermodynamic quantities. But that's of course not because "molecules can travel at a range of speeds", because that's true even for fixed mass. Waleswatcher (talk) 18:57, 14 May 2012 (UTC)
The confusion comes from talking at cross purposes. I don't see anyone confused about the physics, only the clarity is in dispute, such as "because all photons are identical," a confusing notion I was glad to see gone and sorry to see reappear. But if Chjoaygame's ok with that reversion we can declare this horse dead and stop beating it. Interesting to see how long its body remains in the article. ;) --Vaughan Pratt (talk) 22:15, 14 May 2012 (UTC)

## Physical outline

I've drastically reduced the length of the "Physical outline" section, while trying to keep the important physics. The article is very long already, the last thing it needs is a multiple-subsection "physical outline" filled with history and verbose explanations. If other editors feel the current version lacks some crucial piece of physics, add it back in - but let's try to keep this short (it's supposed to be an "outline"), and save the history for the history sections. Waleswatcher (talk) 19:25, 14 May 2012 (UTC)

The new edit by Waleswatcher has many merits, but perhaps it has one flaw. The previous version of this section was felt by several editors to be "pitched at far too high a technical level", so that it might be suitable for a place further into the article. I responded to that by moving the section as it was written to a place further into the article. The version that Waleswatcher has overwritten has some of the features that led to the complaint that the previous version was "pitched at far too high a technical level". I wonder if the same complaint will be made about those features as they appear in Waleswatcher's new overwriting. Also I am not sure that total banishment of history to the history section is appropriate. My "verbose" explanations are intended as explanatory for bears of little brain, and are intended to be less "pitched at a far too high a technical level". Previously during the mighty storm that started last October, I took the history section into a separate article, but that was quickly undone. Vaughan Pratt has suggested a separate history article, and that would indeed shorten the present one and make room for more physics and more explanation for bears of little brain.Chjoaygame (talk) 19:40, 14 May 2012 (UTC)Chjoaygame (talk) 19:49, 14 May 2012 (UTC)
Putting history in a separate article is a good idea. I don't object to having a little bit in the "Physical outline", but only if it serves the purpose of explaining the physics better than modern references and explanations can. I do agree that the level is probably too high, I'll have a stab at that right now. Waleswatcher (talk) 19:56, 14 May 2012 (UTC)
Chjoaygame, as this article's resident expert on what constitutes "chatty writing," would you say the current first paragraph of the article is more or less chatty than my proposed introduction, which previously you complained about as being too "chatty" and "personal" for a Wikipedia article? --Vaughan Pratt (talk) 22:27, 14 May 2012 (UTC)
No comment.Chjoaygame (talk) 23:26, 14 May 2012 (UTC)
The (current section) is still incredibly poorly written. It's like a pseudo-lead, which duplicates the "physics" section, and is mostly a coat rack of various semi-related ideas without any coherence to the presentation. Several claims are unreferenced. I say expand the lead, and just axe this section. What's worth salvaging can be merged in either the "history" or "physics" sections. 00:20, 15 May 2012 (UTC)
I agree, at present it is a rambling wall of text. I think it should be renamed "introduction" and it should orient a new reader to the subject in a few concise paragraphs. Some important points are made, which should be kept, somewhere, but a section of pure text which takes screen after screen to go through is not right. PAR (talk) 02:09, 15 May 2012 (UTC)
As it was, it was shorter than what it replaced. In any case you're right that it was too long, and I've trimmed it further. If you think it's poorly written, edit it or be specific. Waleswatcher (talk) 02:48, 15 May 2012 (UTC)

## Revert

I reverted to this version (previous), the article is much better with general commentary that is not tied to specific forms of Planck's law. The order also made no sense at all. Forms need to be presented before their relation, not the other way around. 00:10, 15 May 2012 (UTC)

On the contrary, this language is both more clear, trims out the (out of place) history, and less technical. Re-reverted. If you have a problem, please be specific: wholesale reverts like that are unconstructive. I do agree that the section is probably still too long - but it's shorter than what it replaced, and can certainly be trimmed further. It's also somewhat introduction-ish, and perhaps moving some of it into the intro is a good idea. Waleswatcher (talk) 02:30, 15 May 2012 (UTC)

I certainly believe an outline and introduction would be far better after the lead than a list of different forms. The lead should outline the topic and summarize the article. It is not an introduction to the subject. The structure of the article should be made right. There is no point starting off with a list of different forms. Dmcq (talk) 18:09, 15 May 2012 (UTC)
People agreed that the article should begin by stating the law, otherwise we're talking about some unknown law. Compare for example with Kirchoff's circuit laws or Biot–Savart law or countless other articles on various laws and equations. But regardless of where the law should be found in the article, this is just awful and redundant with the lead. 00:36, 16 May 2012 (UTC)
If you believe that you should leave the section name at the very least. The structure of an article is important, it is not just a collection of facts. The present title 'Introduction' is fairly anodyne but reasonable. The title in that Biot-Savart law of 'Equations' for the starting section, well that's on about the same level or even worse than 'Different forms'! Kirchoff's circuit laws one of 'Kirchhoff's current law (KCL)' is not too good either because it concentrates on one half only. Dmcq (talk) 09:28, 16 May 2012 (UTC)
That's a bit worrying, Biot-Savart law did start with introduction but that was changed to Equation by this diff. I hope there is not some general idea that one should just cater for people who already know most of what an article is about and only want to see the equations like some table at the back of a book. Dmcq (talk) 09:36, 16 May 2012 (UTC)

Headbomb and Dmcq are both right, the article should begin by stating and explaining the law itself. Aspects of interest to different interest groups are better brought out later -- for example users of the law might not be interested in its physics and vice versa.

Also note that currently Wikipedia has four intimately related articles, Black body, Black body radiation, Thermal radiation, and Planck's law, but the appropriate material for each seems to be in the wrong articles. For example both the physics of black bodies and the properties of black body radiation are being covered much more comprehensively in the Planck's law article than in their own articles.

What hasn't been treated well so far is the difficult point about infinitesimals, which is intrinsic to the law in the sense that the meaning cannot be stated precisely without it and therefore should by the above criterion be in the introduction. Following the example of other editors over the past day or so, I've illustrated this with a rewrite of the introduction that sticks to what the law is and what it means, with the tricky infinitesimal issue explained "as simply as possible but no simpler." As a result the introduction may seem a bit short on other details, but what would you add to it that would not be better located elsewhere? --Vaughan Pratt (talk) 15:42, 16 May 2012 (UTC)

Incidentally I would argue for a sentence or two in the introduction that explains in outline what to do about non-normal radiation, with a reference to the Lambertian section for the full details. Yes or no? --Vaughan Pratt (talk) 16:01, 16 May 2012 (UTC)

### response 1

I think the introduction as you re-wrote it is fine, but it seemed highly redundant with the lead. In particular, what's the point of repeating the equation that's already in the lead in the section immediately following? To me, the point of the introduction is to explain in simple language what this law describes, why it's important, and what it says. People that will benefit most from that probably can't understand equations. As for putting that in a separate article, I wouldn't object to that if it weren't for the fact that we already have four articles on essentially the same topic. Do we really need another? Waleswatcher (talk) 20:34, 16 May 2012 (UTC)

### response 2

I'm a bit surprised this article does not mention the ultraviolet catastrophe. If there is going to be an introduction shouldn't it give just a little bit of the background. Dmcq (talk) 22:57, 16 May 2012 (UTC)

Dare I say it, the history section does mention the ultraviolet catastrophe. There is fairly well supported multiple sourced historical opinion that Planck did not think of 'the ultraviolet catastrophe' until some time after his discovery of his law. The actual phrase was invented by Ehrenfest in 1911. This means that it would be a mistake to say that Planck was motivated by an intention to solve the ultraviolet catastrophe problem. The physics of the catastrophe was, it seems according to historians' opinion, first introduced by Raleigh about six months before Planck made his discovery of his law, but Planck did not register it in his thinking, if he had even read it. He was long accustomed to the Wien cut-off exponential, and after his discovery, when he learnt about the "catastrophe" he said it was due to an unjustified application of the equipartition principle. This is briefly mentioned in the history section. I do not see benefit in further emphasizing it, and in particular I don't think it needs to go into the introduction.Chjoaygame (talk) 23:51, 16 May 2012 (UTC)
I think I must have misspelled the search, yes I knew about that and should have twigged I was doing something silly. Dmcq (talk) 09:40, 17 May 2012 (UTC)
In fact one of the more amusing quotes I've seen is in Planck and others proposing Einstein for the Prussin Academy in 1913 saying "
"That he might sometimes have overshot the target in his speculations, as for example in his light quantum hypothesis, should not be counted against him too much."!
And that long after Planck had realized that energy levels had to be quantized to account for his law. Dmcq (talk) 10:17, 17 May 2012 (UTC)
Yes. According to sources, Planck was a keen supporter of Einstein on special relativity, though not on the hypothesis that light travels as particles. Planck did gradually accept that the generation and perishing of light was quantal, though not the free propagation through space.Chjoaygame (talk) 15:09, 17 May 2012 (UTC)

### response 3

The problem about infinitesimals. An article on Planck's law is not the place for explanations of infinitesimals to people who need such explanations. I think it is a problem created in the mind of VP, as one of the contents of his trojan horse.

What to do about non-normal radiation. It is most easily and naturally dealt with by working, as Planck did, and as many more recent physics texts do, primarily with cavity radiation which is isotropic, as implicitly at least recognized by Balfour Stewart in 1858, who first described it, and by Kirchhoff in 1860. Without the traditional cavity approach, the isotropic character seems almost incomprehensible to a newcomer, no matter how you word it. The 'problem' of non-normal radiation is created gratuitously, by abandoning the traditional cavity approach.Chjoaygame (talk) 00:15, 17 May 2012 (UTC)

The idea (in the article currently as I write) that the law is "nominally" about normal radiation is a misundertanding. I think it is another content of the trojan horse.Chjoaygame (talk) 00:35, 17 May 2012 (UTC)

• (In response to Chjoaygame): How do we know you're not using the physics of thermal radiation as a "trojan horse" for getting that material into the Planck's law article? --Vaughan Pratt (talk) 00:48, 17 May 2012 (UTC)
A trojan horse is a vehicle for getting things in somewhere hidden inside the horse. Your plans on your talk page were my concern there; they seemed to carry a kind of suggestion that what was written by you on your talk page would somehow licence you to put it into the article, with a tacit understanding that it would somehow be privileged against editing by those who had written on your talk page. My putting the material about the cavity approach in the article was direct and open and up-front and not hiding it in something. If you want to say that my advocacy of a physical understanding is a trojan horse, be my guest!Chjoaygame (talk) 04:09, 17 May 2012 (UTC)
• not the place for explanations of infinitesimals My explanation assumes the reader understands infinitesimals, it does not explain them. What it does is to give the meaning of Planck's law in terms of them. One reason why this aspect of Planck's law has given so many editors so much trouble over the years is that there has not to date been a simple and physically meaningful explanation in this article of why infinitesimals are needed for the law and where they enter into the law. --Vaughan Pratt (talk) 00:58, 17 May 2012 (UTC)
The reason for the infinitesimals is that the spectral radiance is a bit more subtle than a simple three-vector, which has just one direction and sense, such as for example a flux density. The physical context is that at any point in a cavity, radiation passes in both senses of every direction through that point. This means that the energy propagating in a specified direction must be separately evaluated for each sense. Also, radiation at a point, in a specified direction, has two possible perpendicular planes of polarization. The quantity spectral radiance is tailored for this physical context, and it is defined in terms of infinitesimals in a way that is obvious when this physical context is made clear.Chjoaygame (talk) 04:09, 17 May 2012 (UTC)
Ok as an exam answer for graders who already understand this stuff and just want to know if you've grokked the material. As an understandable explanation for students who don't it might benefit from a bit more logical coherence and a bit less use of "obvious." Graders appreciate knowing the student finds the material obvious just as much as students hate being told by the instructor that it's obvious. --Vaughan Pratt (talk) 17:05, 30 May 2012 (UTC)
A hemisphere occupies 2π steradian. Spoken like a true theorist. :) If you rely primarily on this fact to predict the spectral radiant exitance into a hemisphere from one side of a planar radiator as observed in practice, you will find the measured value wildly different from your theoretical prediction!
This surely argues for rather than against including the sort of information that would help the reader avoid arriving at this wrong answer. Just because you're not a user of this law yourself doesn't mean the article should not try to provide users with useful information. Your claim that the only worthwhile insights into Planck's law can all be derived from an understanding in terms of cavities, equilibrium, and other physical aspects is like a car mechanic telling a race car driver that an understanding of how cars work tells him everything he needs to know about driving a car.
Along with other laws such as Beer's law, Planck's law plays a fundamental role in climate science, to name just one application area. Someone interested in applications of Planck's law would naturally expect to find the Wikipedia article on Planck's law helpful. I can assure you that the sort of information that your prescriptive format seeks to bar from this article is precisely what is needed in such applications, and is extremely useful in practice, contrary to your claim that it is not.
I've lost count of the number of times you've alluded to your dreaded "trojan horse." As near as I can tell it is some sort of Mordac, a preventer of information. Your approach to discussions on this talk page seems to be to argue against inclusion of material while accusing others of inserting mechanisms for that purpose. I can assure you I have no interest whatsoever in blocking useful information from this article, though I'd be thrilled if the present excessively long article could be split into articles more precisely targeting subject matter for particular interest groups. Your disdain for users of Planck's law for example suggests that Wikipedia would benefit from an article written by and for users. Your novel theory that material such as how to apply a law of physics is off limits to Wikipedia would be news to Jimmy Wales. --Vaughan Pratt (talk) 07:38, 17 May 2012 (UTC)
There, there. I see you even appeal to the authority of Jimmy Wales to support your insulting mis-reads. You so mis-read me that I am better not to reply in detail, lest you mis-read me further, with further appeals to authority.Chjoaygame (talk) 14:58, 17 May 2012 (UTC)

## Charge not mass explains why the number of photons is not conserved

The way the text explains this is misleading. The reason is not the mass, it is because photons don't have a charge. If photons did have a mass, the number would still not be conserved and the chemical potential would still be zero.

Note also that the photon gas will in principle be in thermal equilibrium with a electron positron gas, the latter has a chemical potential of zero. In practice, due to the temperature being much smaller than the electron mass, we can totally ignore this. However, in the early universe the temperature was high enough.

In fact, the current microwave background radiation has a higher temperature than the neutrino background because after the neutrinos decoupled and the neutrino sector had the same temperature as the photon- electron positron sector, the electrons and positrons eventually annihilated and then "reheated" the photon sector. Without that reheating event, the relic neutrino background would today be at the same temperature as the photon background. Count Iblis (talk) 16:07, 15 May 2012 (UTC)

Yeah, I just removed the entire section about that. It's was just plain wrong. Conservation of photons linked to mass? Utter nonsense. 17:56, 15 May 2012 (UTC)
It's not nonsense. If the number of particles isn't conserved, then the chemical potential is zero. The number of photons (or of any other massless particle species) isn't conserved because they can be created and destroyed at arbitrarily small energy cost (the only exception are non-equilibrium situations where photon-number changing processes are rare enough to be ignored). Photons having a charge would not change this, since photon-anti-photon pairs would still be produced. Waleswatcher (talk) 21:28, 15 May 2012 (UTC)
Last week this section was "pitched at far too high a technical level" so that people asked for it to be moved further into the article. This week it is back up front at a technical level so high that the experts do not agree about the basic physics of what it seems to say. Things that are the same are sometimes different.
What does it mean to say that "photons ... can be created and annihilated with arbitrarily small energy cost"?Chjoaygame (talk) 23:40, 15 May 2012 (UTC)
Yes, but the mass doesn't have to be exactly zero. If the mass is significantly smaller than the temperature, then the cavity will fill itself with the particles in question. Take e.g. the electron-positron plasma in thermal equilibrium with the photons in the early universe. This was in thermal equilibrium with the photons and therefore had a chemical potential of zero.
That they can be created at arbitrary small energy costs, leads to the energy density to go to zero as a power of T (as T^4), and not exponentially (it will then be suppressed by a factor of exp(-m c^2/(k T))). Count Iblis (talk) 23:50, 15 May 2012 (UTC)
Additionally, conservation of the number of photons has nothing to do with mass, and everything to do with them being gauge bosons. And the sentence about photons being create at arbitrarily small cost is rather self evident. There is no lower-energy bound for photons. You can create them with 10 eV, 0.1 eV, 0.01 eV, 0.001 eV, etc... This is not the case for e.g. electrons, where you need at least 0.511 MeV per electron to create them. 00:30, 16 May 2012 (UTC)
• @Count Iblis Yes, but the mass doesn't have to be exactly zero. If the mass is significantly smaller than the temperature... Agree - and for massless particles, that's always the case.
@Headbomb Additionally, conservation of the number of photons has nothing to do with mass, and everything to do with them being gauge bosons. Photon number is not conserved under ordinary circumstances primarily because photons are massless, and that fact has nothing to do with photons being gauge bosons. In short, I have no idea what you're trying to say. Waleswatcher (talk) 02:19, 16 May 2012 (UTC)
• Headbomb writes: "the sentence about photons being create at arbitrarily small cost is rather self evident. There is no lower-energy bound for photons." The clause in question was: "photons ... can be created and annihilated with arbitrarily small energy cost". If the meaning of that is 'There is no lower-energy bound for photons' then I think the latter expression is a much clearer expression of it. I think the original expression "photons ... can be created and annihilated with arbitrarily small energy cost" does not have quite such self-evidence of meaning as Headbomb suggests. Perhaps Headbomb's self-evident reading is the author's intended one. Perhaps Waleswatcher will tell us?Chjoaygame (talk) 02:24, 16 May 2012 (UTC)
• Find me one source that argues that zero mass leads to non-conservation of photon numbers. Because it simply doesn't. 02:33, 16 May 2012 (UTC)
Of course it does. Massive stable particles, at temperatures below the mass (for example, any ordinary gas at room temperature), have a conserved particle number because the energy required to create a new particle (namely mc^2) is far greater than the typical thermal energy. That is never the case for massless particles - there is always enough energy to create more such particles, because they can have an arbitrarily small energy. If for instance you start in a state with too few photons for the temperature of the walls/blackbody, more will rapidly be created to fill in the Planck spectrum, where the chemical potential is zero (dF/dN=0). That doesn't happen in a gas of massive particles at "low" temperatures for the reason above. If you want a source, try Huang, p.280. Anyway I've edited the language to remove the offending mention of massless, because I'm trying to improve the article in the age of the universe. Waleswatcher (talk) 02:42, 16 May 2012 (UTC)
In a cavity, photons of wavelength longer than the length of the cavity do not arise. It is of the essence of a cavity that its length is finite. This entails that photons of arbitrarily small energy do not arise in a cavity.Chjoaygame (talk) 12:39, 16 May 2012 (UTC)
Yes, that's true - and as a result, you don't in fact have a Planck distribution in a finite cavity either. Waleswatcher (talk) 12:58, 16 May 2012 (UTC)
• That's simply the conservation of energy, it's got nothing to do with with the conservation (or lack therefore) of particle numbers. Baryon number is conserved, but that's got nothing to do with them having mass. Lepton number is also conserved, and that's a conservation law that's equally as good as the conservation of baryon numbers, despite leptons being much less massive than baryons. As a counterexample, the Z number is not conserved, despite the Z having a rather sizable mass (~91,000 MeV/c2), because the Z is a gauge boson. Likewise for the photon. 14:57, 16 May 2012 (UTC)
• I was wondering why it was mentioned that the photons of arbitrarily small energy could be created?Chjoaygame (talk) 19:39, 16 May 2012 (UTC)
Because in an arbitrarily large cavity - the only kind of cavity in which radiation actually has a Planck distribution - it's true. And as I said, that (plus the fact that photons interact) is the reason why photon number isn't conserved, which then means equilibrium is an extremum of the free energy with respect to photon number, which means the chemical potential for photons is zero (by definition it's dF/dN). If photons had mass, dF/dN would not be zero at temperatures below the mass. Waleswatcher (talk) 20:22, 16 May 2012 (UTC)
This seems like a statement that it is only in an arbitrarily large cavity that photon number isn't conserved?Chjoaygame (talk) 20:32, 16 May 2012 (UTC)
No, it's the statement that in a finite cavity, photon number will be conserved if the temperature is below the inverse size of the cavity (it will be zero, or exponentially close to it, and independent of T as long as T<<1/L). Anyway, why are we still talking about this? Is there something in the current article you object to? Waleswatcher (talk) 20:36, 16 May 2012 (UTC)
The article (currently as I write!!) says that "The total number of photons is not conserved." The context is of a finite cavity with perfectly reflecting walls containing light but not matter.
Now having learnt the sense in which you meant it, I was wanting to know why (currently as I write!!) in the further-in physics section of the article you state that "photons ... can be created and annihilated with arbitrarily small energy cost"?
You explicate your response with a statement about a finite cavity in which photon number will be conserved.
I was asking about the other case, in which photon number will not be conserved. Your initial response above was about an arbitrarily large cavity, which I suppose is not a finite cavity?
What about this case of interest in the article, in which photon numbers will not be conserved, in a finite cavity? In this case is it so that "photons ... can be created and annihilated with arbitrarily small energy cost" in the sense now apparent as the meaning of that statement?Chjoaygame (talk) 22:47, 16 May 2012 (UTC)
Even in a cavity with perfectly reflecting walls and no matter, on some possibly very long time-scales photons interact and their number is not conserved, as we've already discussed at length. In a more ordinary (non-perfectly reflecting walls, meter sized, room temperature) cavity, photon number is not conserved by ordinary interactions with the walls, and the energy gaps due to the finite size are negligible (the thermal energy is much larger than the minimum energy). Waleswatcher (talk) 02:10, 17 May 2012 (UTC)
I do not see in your immediately above response the reason for the presence in the article of the statement "photons ... can be created and annihilated with arbitrarily small energy cost". My post just above was asking "What about this case of interest in the article, in which photon numbers will not be conserved, in a finite cavity? In this case, is it so that "photons ... can be created and annihilated with arbitrarily small energy cost", in the sense now apparent as the meaning of that statement?"Chjoaygame (talk) 03:11, 17 May 2012 (UTC)
I removed that statement from the introduction; if it's elsewhere in the article I didn't put it there. The meaning of "photons ... can be created and annihilated with arbitrarily small energy cost" is, you give me an energy E>0, no matter how small, and I give you a cavity in which photons of energy E or less can be created and annihilated. Simple enough, I just choose the cavity size L > 1/E. Waleswatcher (talk) 11:55, 17 May 2012 (UTC)
In response to suggestions from Count Iblis and Vaughan Pratt that the first "too technical" version of the physics outline be moved deeper into the article, I moved it intact to the physics section, as a new sub-section, in one step of editing.
In a second step I merged it with the introductory paragraph of that section, taking care as well as I could to leave the text just as it was written, except for trivial changes to make a grammatically smooth merge. That is how it got into the physics section, which I have not since modified, so far as I can recall. It seems you did not notice that.
Perhaps you would be kind enough to do what you think necessary to the version that is in the introductory paragraphs of the physics section of the article?
There are significant differences between (a) the approach to equilibrium in a finite cavity with perfectly reflecting walls and no material content, but only radiative content, and (b) the approach to equilibrium in a finite cavity with walls that are opaque to all wavelengths and not perfectly reflective to any wavelength or with perfectly reflecting walls and containing some black matter. It seems to me it would be a good idea to make this explicit and to give an account of it in the physics section. The physics section already contains an account of the Einstein A and B coefficient theory for the case of the presence of matter. You might be the ideal editor to write a section, about the predictions of the quantum field theory and how they will soon be verified by experiment, for case (a)?Chjoaygame (talk) 14:50, 17 May 2012 (UTC)

Since photon number and chemical potential are conjugate, the uncertainty principle applies and therefore any knowledge of photon number entails some imprecision in chemical potential. In particular when you picture countably many photons traveling through the space between two media you cannot at the same time claim their chemical potential is precisely zero.

That's not to say that the idea of zero chemical potential for photons is false, but just that if you believe it to be true then you must also accept that there is no well-defined number at all of the photons so traveling. In mathematics "uncountable" means "too many count," as with the uncountably many reals. In physics "uncountable" means "can't tell how many" in the sense of Heisenberg.

But if the number is completely undefined, how can an EM wave be meaningfully viewed as comprised of photons as individual particles?

Conversely if you describe an energy level transition for an electron in terms of absorbing or emitting exactly one photon, you cannot know the chemical potential of that photon at all because you know exactly how many photons are participating.

The only way I know of making sense of this situation is to ascribe zero chemical potential to freely traveling EM radiation and not speak of particular photons in it, and to count photons participating in electron energy transitions without attempting to estimate their chemical potential at the time. In plainer language, free photons don't exist in the sense that they can be distinguished counted, only bound ones, and the bound variety cannot have a zero chemical potential.

If you have a better story I'm all ears. :) --Vaughan Pratt (talk) 09:29, 17 May 2012 (UTC)

In any canonical ensemble, you want to sum over all states with weight exp(-beta E), possibly subject to some constraints. For a photon gas, that's a sum over all mode occupation numbers n, where E for a given mode is proportional to n. The chemical potential is zero because you've got -beta*E*n, not -beta(E+mu)n for some mu=/=0. Note that for a gas of massive particles, if "E" is the kinetic energy, "mu" will be the rest mass energy mc^2 (and one may need to include an ensemble of antiparticles if there's a conserved charge). So that's one way to see why mu is zero for photons, as I tried to explain above. As for uncertainties, for a photon gas there is of course uncertainty in the total number <N>: <N^2>-<N>^2>0. Waleswatcher (talk) 14:24, 17 May 2012 (UTC)

## Why the appendix written by me has been deleted?

Dear, Mr. Chjoaygame has deleted the appendix written by me. I would like to the relation between the special value of Riemann zeta function at 4 and Planck's law, which opened the door to Quantum Theory in 1900. Is not it very important historical fact? In general some quantum physical phenomena is used to relate zeta function and enumerate some particles as one, two, three,,,. And their statistics constructs dynamics, i.e. quantum statistics, I think. Then I think we need at least this important fact, not new discovery, as a appendix. If no answer in three days, I will resume it. Otherwise please something new appropriate title, If Mr. Chjoaygame's reason to delete it is absolutely different context. But 6 or 7 years ago the same appendix written as mime had been published, but someone had deleted it. In Japanese version it remains now the article at that time. I hope your presentation to nice direction. talk) 23:35 30 July 2012 (UTC)Enyokoyama (talk) 22:52, 30 July 2012 (UTC)

• The proof is very long-winded. If we generalize the integral to
${\displaystyle f(s,z)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}/z-1}}\,dx}$
then a Taylor series of the integrand about z gives
${\displaystyle f(s,z)=\int _{0}^{\infty }\left(\sum _{n=1}^{\infty }x^{s-1}e^{-nx}z^{n}\right)\,dx}$
which can be integrated term-by-term to yield
${\displaystyle f(s,z)=\Gamma (s)\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{s}}}}$
so that ${\displaystyle f(4,1)=\Gamma (4)\zeta (4)={\frac {\pi ^{4}}{15}}}$
Even this short proof might be too much for the main body of the article. The subject of the deleted appendix is mathematical detail, not about Planck's Law. Such a proof, unless very short, belongs in a mathematics article, or in a footnote to this article. The above proof is taken from the polylogarithm article, but it takes some reading of that article to find the right approach for this specific case. I am in favor of a footnote outlining the proof, perhaps in as much detail as above, and referring the reader to the polylogarithm article. PAR (talk) 14:45, 31 July 2012 (UTC)
• I (chjoaygame) deleted the appendix, as I said in my cover note, because it was very badly written and it was hardly feasible to repair it piecemeal. If it is to be restored, the restoring author needs to thoroughly re-construct and re-write it. A simple sample of the poor writing is that Planck was said to have published in 2000, not 1900 as he did. This is only a simple sample, and by itself it could have been repaired piecemeal. But it is very far from being by itself; it is just a simple sample; the whole appendix was very poorly constructed and very poorly written. I cannot comment on the Japanese version since regrettably I do not know Japanese.Chjoaygame (talk) 16:56, 31 July 2012 (UTC)

## Significance of "Different Forms".

If you look at the different formulae for the different forms (eg whether the left hand side is defined as intensity, or number of photons, or energy, etc, etc), one significant thing is that for any one temperature, each of the different forms gives a slightly different wavelength for the maximum value. This is important in terms of what physical behaviour you are trying to describe. And that's about all I remember about this from physics lectures in 1974. 143.238.84.30 (talk) 11:15, 19 October 2012 (UTC)

## Waleswatcher's unsourced own research edit

Waleswatcher has undone the main features of my edit. He is making it up as he goes along, not working from sources. For example, he removes the qualification "rigid" that the source, Planck, routinely includes. Waleswatcher's reason in the cover note is that a rigid cavity cannot contain wavelengths longer than its own size. This is not to be found in the source as a reason concerning the rigidity of the walls, and is Walewatcher's own research; we can tell this because he offers his reason as "for one thing". Waleswatcher's idea seems to be that somehow non-rigid walls will be just as suitable for the statement of the law, but no source is offered by Waleswatcher for this innovation of his.

Waleswatcher has removed Planck's concern to point out that the distribution is stable, and omitted Planck's point that there is only one stable distribution, providing the reason for the word 'unique'. Waleswatcher has invented his own research for the meaning of 'unique' here, as a repetition of his idea that the only requirement on the material of the walls is opacity (never mind that he omits the requirement stated by Planck, that the walls also be rigid). It may be useful in the article to explicate this further, by adding explicitly that the chemical composition of the walls is immaterial, but this may be more than is needed if the reader can work it out from the stated requirements of opacity and rigidity. The statement that the walls are opaque and rigid is more precise than Walewatcher's that they are 'sealed', especially since Waleswatcher thinks the walls should not be specified as rigid. Movement of the walls has its important place in the thermodynamic arguments that were used on the way to Planck's law.

Waleswatcher has introduced his unsourced editorial opinion that the Planck law is a useful approximation for ordinary objects, whatever an ordinary object might be. Those who worked for years on finding the empirical data for Planck's law would have had different views about how easy it is to approximate a black body by an ordinary object, whatever that might be. This opinion is not suitable for this place in the article, which is introducing the main features of the law; perhaps Waleswatcher made it explicit here to justify his restoration of the vague wording of the initial sentence of the section.

The language of Waleswatcher's wording "Planck's law does not depend on the direction of emission ..." is poor.

Waleswatcher's undoing of my edit was a backward step, devoid of redeeming improvement. I will not right now undo his undoings, because I know the likely outcome of my doing so.Chjoaygame (talk) 18:21, 2 December 2012 (UTC)

### response of PAR

Why are rigid walls necessary? If I blow up a balloon that is opaque except for a small transparent window and let it equilibrate, then that window will emit practically black body radiation, yet the walls are not rigid. PAR (talk) 18:35, 2 December 2012 (UTC)

The source is Planck's well recognized text. Planck says the walls should be rigid. It is important physics that when the walls move, the movement affects the spectrum. At least something needs to be said along those lines. Your balloon has relatively rigid walls, which got that way when you blew it up. Not perfectly rigid, I agree, but you did feel the need to blow it up to give it some shape.Chjoaygame (talk) 19:52, 2 December 2012 (UTC)
Ok, the walls need to be stationary, but not necessarily rigid. The thermodynamic parameters of a system in equilibrium are unchanging in time. When you say that Planck's law is an equilibrium distribution, that includes the fact that the volume is not changing. It might be nice to mention that the walls are not moving, but not mandatory. PAR (talk) 00:31, 3 December 2012 (UTC)

### response of Waleswatcher

Chjoaygame, calm down. Your hysteria doesn't help. First off, Planck is not really the main source for this article. Planck is primary, and wiki is supposed to rely on secondary sources (of which there are plenty). No finite volume cavity can contain an exact Planck spectrum, for the obvious reason I gave in the edit summary (long wavelengths don't fit in the cavity). Either that subtlety should be explained carefully, or we should avoid words like "rigid" that pretend it's possible. Even "opaque" was originally your word, if I recall correctly. I'd be in favor of deleting it for that same reason. As for ordinary objects, Planck's law is used by engineers, physicists, astronomers, chemists, and many other people routinely as an approximation to the spectrum emitted by various objects. There are thousands of sources for that; I'd be happy to add some if you think it's so important. Obviously, an article on Planck's law needs to clearly and prominently point out how useful and important it is, rather than just being some kind of theoretical idealization that only applies to "black bodies". Waleswatcher (talk) 19:12, 2 December 2012 (UTC)

Oh and as for "stable", it's an obscure word to most, and doesn't fully capture the most important feature. The physics is that the Planck distribution is the equilibrium distribution for radiation, and it's unique once the temperature is specified. It is indeed stable, but it's much more than that, it's an essentially universal attractor. All distributions of photons will tend to Planck, as a consequence of the second law of thermodynamics. I'll add something like that to the article. Waleswatcher (talk) 19:16, 2 December 2012 (UTC)

Waleswatcher, thank you for calling me hysterical. It tells us something about your needs. You need to do that because you can see that my arguments are sound and hard to refute and so you need some rhetorical trick, some spin, to disparage them in order to support your position.
Planck is the only cited source in this section. Planck is a recognized authority, recommended by present-day writers, for example Goody & Yung (1961/1989), p.64. Planck's text is a fair candidate for being a secondary source, since he was a towering authority for many years in his day, and his text considers the work of other writers since his original publications, which are indeed primary, but are not the cited source here. You cite no source, and at the same time get Wikilawyerish about a supposed difference between others' primary or secondary sourcing. I do not think you are likely to produce better sources.
Contrary to your claim, 'stable' is not an obscure word to most; I hear it every day on the news, which tells me that someone ill in hospital is in a stable condition. I am not endorsing how the news writer uses the word. Planck took care to point out the stability of the thermodynamic equilibrium distribution. You try to condescend on this, by telling us that the distribution is much more than stable, that it is an essentially universal attractor. Planck used the second law to establish the stability. You want to cite the second law, but to hide Planck's priority about it.
'Opaque' is not my word. It's Planck's and is repeated in many reliable texts.
I repeat, your meaning for 'unique' in this context is your own invention, contrary to the source.
The word rigid is not meant to imply, as you claim, that the cavity can contain wavelengths longer than its size. It is meant to say that the walls should not move. You claim that it is meant to imply that the cavity can contain wavelengths longer than its size is mere confabulation or prevarication; you are making it up as you go along.
If you want to put something in about Planck's law being used as an approximation for non-black radiators, of course you are free to do so. But to do so right at the initial statement of the law is an unhelpful distraction.Chjoaygame (talk) 20:23, 2 December 2012 (UTC)
If there's something in the text of the article that you think is unsourced or incorrect or both, identify the specific passage and we will discuss it. Waleswatcher (talk) 21:27, 2 December 2012 (UTC)

## Chjoaygame's latest revision

I undid this revision by Chjoaygame, there is absolutely no requirement for perfectly reflecting walls, it doesn't exist and, if it did, there would be no change in the radiation in the cavity because the walls, being defined as perfectly reflecting neither absorb nor emit radiation and thus have no thermal coupling with the radiation in the cavity. This (impossible) situation means relation between the cavity walls and the radiation is undefined. --Damorbel (talk) 09:38, 4 December 2012 (UTC)

Damorbel's complaint that a cavity with perfectly reflecting walls does not really exist in nature was anticipated by my post, which included the word "theoretically", ignored by Damorbel. It is true that theoretically perfectly reflecting walls are not necessary for a restoration of a Planck distribution, but with walls that are not theoretically perfectly reflecting, the sentence that I posted would be of little interest, because the work would not be done adiabatically. The sentence that I posted is of interest only for work done adiabatically.
because the work would not be done adiabatically. You omit the fact that "perfectly reflecting walls" do not interact with, i.e. they can neither emit nor absorb, radiation; that is why the system is undefined. Cavities that reflect perfectly contain only the radiation present when the cavity was made. Let us imagine the cavity sprang into existance from nothing, it would not contain any radiation, nor could any radiation enter it ("perfectly reflecting walls" - you see). --Damorbel (talk) 10:43, 4 December 2012 (UTC)
Reflection in a moving perfectly reflecting wall is an interaction. It changes the wavelength of the light.Chjoaygame (talk) 11:35, 4 December 2012 (UTC)
As the theoretically perfectly reflecting walls are moved, the light rays reflected from them suffer changes of frequency because of the motion of the walls. These changes of frequency reconstruct a new Planck distribution just precisely suited to the new amount of energy in the body of radiation due to the work done. This argument was used, in a slightly different form, by Wein Wien during the saga of the discovery of Planck's law. In his undoing of my edit, Damorbel is mistaken in physics.Chjoaygame (talk) 10:06, 4 December 2012 (UTC)
Wien? Wein? It helps if you remember - "Drink wein in Wien - not Wien in wein!" (Yuk!) --Damorbel (talk) 10:47, 4 December 2012 (UTC)
It helps if you read the literature.Chjoaygame (talk) 11:35, 4 December 2012 (UTC)
Reflection in a moving perfectly reflecting wall is an interaction. It changes the wavelength of the light
I'm not at all sure that this "changes the wavelength"; how does it do this? There may be a Doppler effect from a moving mirror. And thinking of the change of volume the density of photons in the cavity enclosed by the mirror(s) will increase but, without any absorbing material in the cavity, this is not going to change the temperature.
I think the problem of defining the initial conditions is not yet solved. --Damorbel (talk) 13:34, 4 December 2012 (UTC)
You could always try reading the source for yourself.Chjoaygame (talk) 19:18, 4 December 2012 (UTC)
Try page 69, second paragraph, Chapter III of Part II, headed Wien's Displacement Law, of Planck, M. (1914).Chjoaygame (talk) 16:07, 5 December 2012 (UTC)

## Waleswatcher edits to introduction

I'm not going to engage in an edit war. If other editors wish to support one version or the other or a new one, let them make that edit and I will support what I think is best. I think the present version is obviously in error. If no other editors wish to make any changes, then the present version stands, as far as I am concerned. PAR (talk) 13:30, 12 December 2012 (UTC)

To me, PAR seems right to say that the former version was obviously in error, and needed remedy. The replacement version by Waleswatcher was illogical, and needed remedy. The emissivity makes the description as accurate for non-black bodies as the law itself is for black bodies. The ideal, maximal, character of Planckian radiation deserves explicit statement up front.Chjoaygame (talk) 13:41, 12 December 2012 (UTC)
What specifically do you think is in error? Regarding emissivity, it's still approximate as is written, because ordinary objects are never in perfect thermal equilibrium. Waleswatcher (talk) 13:43, 12 December 2012 (UTC)
As for the "maximal" character of Planck radiation, I don't object to mentioning it, but what was written was wrong. It asserted that "Real bodies in general, and especially gases, emit less than black bodies." That's manifest nonsense - a flashlight emits much more visible radiation (and much more total EM energy) than a perfect blackbody at the same temperature. And gases can emit just as much as anything else, it simply depends on the optical depth. So if such a statement is going to be made, it needs to be made much more carefully, and I don't agree that with all the necessary caveats it belongs at the very beginning of the intro. Instead, it would be better placed further down. Waleswatcher (talk) 13:49, 12 December 2012 (UTC)
Two points:
1. "Planck's law describes the electromagnetic energy radiated by objects at a given temperature" is simply wrong. Subsequent statements do not make it right.
2. The statement that ""Real bodies in general, and especially gases, emit less than black bodies." is incomplete, but not wrong. The element of equilibrium is missing. Real systems at equilibrium emit the same or less radiation than a black body at equilibrium. In other words, emissivity is never greater than unity. The filament in a flashlight bulb emits less radiation than a black body at that same temperature, always. A gas at equilibrium never emits more than a black body at equilibrium. "Emit" does not count radiation passing through the gas - optical depth has nothing to do with it. If I shine a flashlight through the atmosphere, the atmosphere is not "emitting" that light. PAR (talk) 13:58, 12 December 2012 (UTC)
You say the first statement is "simply wrong", while the second is "incomplete, but not wrong". And yet the first statement is simply incomplete in precisely the same sense as the second - both require additional caveats. As for your specific points, let's take them one by one.
*Real systems at equilibrium emit the same or less radiation than a black body at equilibrium. Real systems are never in equilibrium.
*The filament in a flashlight bulb emits less radiation than a black body at that same temperature, always. My flashlight doesn't have a filament - it uses LEDs, which emit a light that is nowhere near thermal. The total energy in radiation it emits is far larger than the Planck rate associated to its average temperature.
*optical depth has nothing to do with it. Sorry, but that's flat-out wrong. Optical depth tells us how close the gas is to a perfect absorber, which as always determines the emissivity. The sun is a collection of gas, and yet it's a pretty good black body. Why? Because it's optically thick. The most perfect black body every observed is the cosmic microwave background, which was emitted by.... you guessed it, gas. Waleswatcher (talk) 22:36, 12 December 2012 (UTC)
Lets take your points one by one:
* You say the first statement is "simply wrong", while the second is "incomplete, but not wrong". And yet the first statement is simply incomplete in precisely the same sense as the second - both require additional caveats.
No - in the statement "Two plus two is five, but is exactly equal to four", the phrase "but is exactly equal to four" is not an "additional caveat" it is a contradiction of the phrase "Two plus two is five", which is simply wrong. "Planck's law describes the electromagnetic energy radiated by objects at a given temperature" is simply wrong, and no caveats will fix it.
*Real systems at equilibrium emit the same or less radiation than a black body at equilibrium. Real systems are never in equilibrium.
Fair enough, but you know what I am trying to say. If I make a statement that is less than perfectly rigorous and you know what I am trying to say, please correct the lack of rigor if it bothers you, and address the statement, rather than nit-picking it and discarding it. Let me amend the statement to say that "Real systems at (practical) equilibrium emit the same or less radiation than a black body at equilibrium at that same temperature."
*The filament in a flashlight bulb emits less radiation than a black body at that same temperature, always. My flashlight doesn't have a filament - it uses LEDs, which emit a light that is nowhere near thermal. The total energy in radiation it emits is far larger than the Planck rate associated to its average temperature.
This last sentence is in total contradiction to the first sentence you wrote in the introduction, namely "Planck's law describes the electromagnetic energy radiated by objects at a given temperature". But never mind. You say that the light is "nowhere near thermal". That is because it is being radiated by a body that is not in thermal equilibrium. We are not talking about systems that are far from equilibrium, we are talking about systems that are at least in LTE, where the massive particles are in thermal equilibrium. Again, the concept of equilibrium is vital and cannot be ignored.
*optical depth has nothing to do with it. Sorry, but that's flat-out wrong. Optical depth tells us how close the gas is to a perfect absorber, which as always determines the emissivity. The sun is a collection of gas, and yet it's a pretty good black body. Why? Because it's optically thick. The most perfect black body every observed is the cosmic microwave background, which was emitted by.... you guessed it, gas.
This is not the issue. I agree, the sentence is vague and could be better written. The point that is trying to be made is that small volume element of a gas in which the gas particles are in practical equilibrium at a particular temperature will not emit more radiation in a particular direction than a black body element at that same temperature would emit. This statement makes no reference to optical depth. PAR (talk) 16:12, 13 December 2012 (UTC)
If, in order to justify a story of approximation, one wants to talk about non-black bodies out of thermal equilibrium, one should identify that as the ground of the approximation. Therefore new version as written by Waleswatcher is logically faulty. It is also poorly written, but Waleswatcher has a track record of violently insisting that his edits prevail word for word, and so I will not revert his new version right now, in order to avoid an edit war. The rate of edits here and now is hard to keep up with in real time. The Wikipedia is the loser.Chjoaygame (talk) 14:06, 12 December 2012 (UTC)
What precisely is "logically faulty"? As usual, you make assertions like that without actually identifying anything specific. Waleswatcher (talk) 22:25, 12 December 2012 (UTC)
I wrote: "If, in order to justify a story of approximation, one wants to talk about non-black bodies out of thermal equilibrium, one should identify that as the ground of the approximation. Therefore new version as written by Waleswatcher is logically faulty." Thinking them obviously implicit, I did not actually write also the words 'the ground of the approximation is not stated,' but you complain that I am prolix, and I thought that you would be able to work them out for yourself. Apparently not. My word "Therefore" was not enough to guide you back to the previous sentence. Well, here it is: The precise logical fault is that the ground of the approximation is not identified.
Sad to say, you are not engaging with the other note that I made there: "Waleswatcher has a track record of violently insisting that his edits prevail word for word." You are here now making word for word attacks on the edits of other editors, with a view to restoring your own edits word for word. Much of what you write here is trivial in the mediaeval sense of the word. Broadly speaking, you are now making the edit process unduly difficult and futilely time-consuming by your nit-picking, which is aimed not at construction, but rather is aimed at obstruction.Chjoaygame (talk) 00:07, 13 December 2012 (UTC)
Do you see what it says at the top of this page, Chjoaygame? "Be polite, assume good faith, avoid personal attacks". As usual, you have followed none of that advice. As for nitpicking, if only you could appreciate the irony of you accusing me of that. Waleswatcher (talk) 02:45, 13 December 2012 (UTC)
One starts with assumptions, and tests them against experience, and holds to them when they accord with it.Chjoaygame (talk) 05:19, 13 December 2012 (UTC)