Wikipedia:Scientific peer review/Equipartition theorem

Equipartition theorem

Hi, this is a pretty important classical physics topic. Its failures helped to spawn quantum mechanics and even now it's pretty useful. Please let me know what you all think of the presentation, and derivations. I'll try to flesh out the article titles and page numbers for the 19th century journal references but, if you happen to know of some already, I'd be very grateful. I'd like to bring this to Featured Article status in the near future. Thanks for your time and thoughtful reviews, Willow 20:08, 29 March 2007 (UTC)

Cryptic C62

I'll read through it and comment as I go.

• "Similar examples could be cited for every ..." this sentence is very unencyclopedic and unnecessary. Fixed
• What are "first principles" ? Fixed
• Can theorems form alliances? "The history of the equipartition theorem is closely allied with that of specific heats" Fixed
• There are alot of "unknowns" and question marks in the references. I'll work on this; it's hard to track down 19th-century references! :(
• You have a lot of high-quality sources. However, these all seem to be used in the History section. Make sure the applications and equations are well-referenced. The entire Derivations section is missing citations. Fixed
• "Pair potential" is red-linked. Either whip up an article for it or give a brief explanation here, as it's not a well-known concept. Fixed
• I doubt this is even possible, but if you can find or create a visual for this article, it would definitely help. It is very dense reading, as would be expected of physics theorems. Two figures; any suggestions for more?

On the positive side, this information seems very comprehensive and well-organized. Keep at it! --Cryptic C62 · Talk 03:31, 31 March 2007 (UTC)

Thank you very much, Cryptic! :) I'll try to address your concerns. Willow 16:16, 2 April 2007 (UTC)

Here's a few more little fixies for the intro:

• "Indeed, the failure of the equipartition theorem to predict the specific heats of solids and diatomic gases was the first hint to physicists of the 19th century that classical physics was incorrect and that a new physics — now understood as quantum physics — was needed." I have two problems with this sentence:

• It's very long, compounded by the fact that it doesn't have commas. Consider either breaking it apart or removing some of less important phrases, such as "to physicists of the 19th century."
• The word "hint" seems like an odd choice. What about "sign" or "evidence"?

• "to predict the specific heats of solids and diatomic gases" is used twice within the same paragraph.
• "For example, the ideal gas law can be derived from equipartition. So can the Dulong-Petit law, which describes the specific heat of all solids at high temperatures." that can probably be schlorbed together. I like "The ideal gas law, for example, can be derived from equipartition, as can the Dulong-Petit law." or "Both the ideal gas law and the Dulong-Petit law can be derived from equipartition."
• Why is "quadratically" italicized?

These may seem overly nitpicky, but phrasing and flow is especially important for the introduction. Many a time have I read the introduction to a theory article and gotten discouraged because of how dense it was. An easy-to-read intro will encourage more people to read the entire article. --Cryptic C62 · Talk 01:26, 14 April 2007 (UTC)

Hi Cryptic, thanks for your excellent notes! I've tried to fix up the lead as you recommend, and made some further changes as well. Please let me know what you think! :) Willow 23:36, 14 April 2007 (UTC)

Good revisions. However, I noticed the section title "Failure in the quantum regime". Regime seems like an odd word choice. While it can be defined as a ruling or prevailing system, the dominant definition and the most common connotation is a dictatorship. I'm fairly certain that quantum mechanics were not a precursor to Saddam Hussein. :P --Cryptic C62 · Talk 03:26, 15 April 2007 (UTC)

I'm not sure what I was thinking; it does sound really poetic, doesn't it? I must've heard someone say that somewhere, and it stuck to my brain like Velcro. ;) Hopefully, it reads better now; thanks for the tips! :) Willow 21:11, 18 April 2007 (UTC)

Google finds over 90,000 hits for "quantum regime" and over 73,000 for "classical regime" -- it's very much a standard phrase. Jheald 14:57, 23 April 2007 (UTC)

Google is not an all-knowing entity. That's Wikipedia. Even if we are using Google as a proving ground, 90,000 pales in comparison to the 5,850,000 hits for "quantum mechanics." To the average user, "regime" will incorrectly imply dictatorship. --Cryptic C62 · Talk 18:11, 23 April 2007 (UTC)

Hey, I'm back. I added redirects at Law of equipartition and Equipartition theory. I also noticed the Maxwell Boltzmann image uses nonstandard isotope notation, He-4. They should be either Helium-4 or ⁴He. Try recreating the image, or contact its author, Pdbailey. --Cryptic C62 · Talk 01:46, 23 April 2007 (UTC)

Oops. I did it myself. --Cryptic C62 · Talk 00:15, 24 April 2007 (UTC)

Jheald

A couple of comments, not a full review (more comprehensive comments, maybe, in a day or two)

• Something the article does is link to Hamiltonian quite early on with relatively little explanation. Despite my first instincts, I quite like that the article does this, because it's a useful set-up for the relativistic gas example, which is a very useful corrective to the idea that the energy is always ½kT per degree of freedom. But the idea of a Hamiltonian is quite an advanced concept compared to the entry level for a lot of the potential audience for this article. And Wikipedia isn't giving you much help at the moment. At the moment the link of the page is pointing to the Hamiltonian disambig page - which probably isn't going to help, if a reader has never heard of the word before. But the articles WP does currently have, eg Hamiltonian system, Hamiltonian mechanics, Hamilton's principle and Hamilton's equations don't currently altogether help either -- I fear that none of them has an introduction/summary which is pitched remotely simply enough for the entry-level of people coming to this article; and they could use being knocked together big-time. The one which probably most ought to be fixed up as a potential entry-point is Hamiltonian mechanics, which is supposed to be the category lead for Category:Hamiltonian mechanics. An introductory paragraph there, after the contents, glossing (but not proving) some of the results set out in Hamilton's equations, in particular H=T+V and the form of the equations of motion, might be the way to go. In fact the best solution would probably be to merge the two pages together outright, with most of the content now at Hamilton's equations coming in first, and the material now at Hamiltonian mechanics coming in as things get more sophisticated.

You're definitely right, although I had hoped to avoid all the work. When I first started out here, I also encountered some difficulties with a few pure-math Wikipedians over my overly basic description of the Hamilton-Jacobi equation; I kind of dread having to wrangle with them again. :(

Okay, I've added the merge tags, as an indicative first step. Let's see whether anybody wails! Jheald 14:14, 3 April 2007 (UTC)

• Pictures. It might be nice to show a pic of something like the Maxwellian distribution of 1d kinetic energies, with 1/2 kT marked, to show how much energy spread there can be in a single quadratic co-ordinate; and then a pic of the energy per d.o.f. for 3d and say 12d, to show how the energy per d.o.f. becomes very sharply defined for systems of more degrees of freedom as the Central Limit Theorem kicks in. Would pics for the corresponding relativistic gas show nice qualitative differences ?
• Pics showing the effects of frozen-out d.o.f.s due to energy-level spacing might be nice, too - showing how this makes an impact even before you put in Bose-Einstein or Fermi-Dirac effects. The nice thing about pics is they allow you to talk qualitatively about the effects, while being able to defer the quantitative details.

Maybe there's a way of combining these two figures to show how equipartition fails when you go from continuous to quantized energy levels. I'll brood on it for a few days.

• For the very simple quadratic case, would it be an idea to work through the maths starting with Boltzmann factors from the canonical distribution? Without prejudice to the more detailed and general derivations still being considered later.

Just some thoughts. Jheald 20:18, 2 April 2007 (UTC)

Yes, maybe. I'll try to add that and let's see how it flies! :) Thanks for your comments! Willow 11:55, 3 April 2007 (UTC)

Gnixon

Hi Willow. Thanks for pointing me to this informative article. It contains a lot of great information and some useful equations. However, I'm very concerned that the article is having a hard time deciding on the level of presentation. If it's going to be technical in nature, the theorem and its derivations need to be right up front. If it's going to be less technical, it can't discuss the theorem in such general terms. I was frustrated reading the first few sections because, first, the theorem and its consequences were discussed in detail without actually stating the theorem, and second, the theorem was stated in a very general form (in the Hamiltonian formalism but without explaining it) but derivations were pushed to the end. A start at improving things, which I think another reviewer suggested, would be to first present the theory in a very simple form, perhaps only explaining the E=N/2*kT result and alluding to an elementary derivation. Later in the article the theorem could be generalized. At its current level, I think the history section needs to come after the more general explanation. It'd be nice to discuss the history without such reliance on the general formulation. I'm reminded of the term "weasel words," usually used to refer to POV issues where awkward language is used to avoid saying something offensive; here, I think there are weasel words used to allude to highly technical concepts without defining them. Finally, as I understand things---disclaimer: I'm no thermodynamicist, and it's been awhile since I studied the subject---the equipartition theorem only "breaks down" in the quantum regime due to ergodicity. I suspect the quantum breakdown should be described as a failure of ergodicity. That certainly seems to be the case for the ultraviolet catastrophe, unless there's something I don't understand. Also, the description of Einstein proposing quantum stuff way ahead of everyone else seems to be an exaggeration of the history as I understand it, but I'm no expert there, either. In summary, I can't support this article as a featured or even good article unless some fundamental problems are addressed. Again, though, there's lots of good information here, and I hope it can be shaped into a good article. Sorry for this long, rambling paragraph. Perhaps I can explain my thoughts better if this becomes a dialogue. Best wishes, Gnixon 02:36, 12 April 2007 (UTC)

Hi Gnixon,

I didn't notice your reply until just now, sorry!

• I agree about the difficulty of presentation. I originally presented everything in its most general form immediately but, as you see above, that was not well received. On the other hand, though, I'm not willing to say (incorrectly) that "equipartition is the equal division of energy kT/2 into all possible modes." It's difficult, isn't it? I'll have to brood on it some more. :)

Yeah, it looked like the article reflected a history on this point. By the way, let me apologize up front if I'm getting any of the theory or history wrong---you're clearly much more familiar with both than I am, so you may have to take some things I say with a grain of salt. Deciding on the level(s?) of the article and its overall structure will probably be the hardest part, and the part you'll get the least help on from anyone else. Unless this article is only going to be a short sub-article for stat mech (in which case I'd do the heavy math right up front at the level of the parent article, keeping everything short and not worrying much about the history, etc.), I think you've got to find a way to introduce things simply, formally but without the full aparatus. Gnixon 00:36, 14 April 2007 (UTC)

• Ergodicity is not a problem for the ultraviolet catastrophe, since all modes may be populated, no? Energy can be freely exchanged between the modes and a set of oscillators that emit and absorb the radiation. Equipartition truly breaks down in quantum systems, because the energy spectrum is not continuous. As you see from the calculation, the mean energy in a single oscillator is not always kT/2, as predicted by equipartition.

I guess I meant it in the sense that certain values of energy cannot be exchanged in the quantum regime, so we can't "freely exchange" energy with low-lying states. That's probably an abuse on my part of the concept of ergodicity. After a bit more reading, I'm still not able to express why classical stat mech had an ultraviolet catastrophe but the h*nu hypothesis solved things. Some famous physicist (Einstein, right?) said that if you can't explain something in simple terms, you probably don't understand it. I clearly don't fully understand the classical blackbody derivations---I was always a little hazy on classical stat mech. It's moderately embarrassing to me since I apply Planck's law frequently in my work.

Some more criticism, which I hope is constructive. I'm not that bad at physics, but I find that as I read the article, there are a lot of things I don't immediately follow. I'm sure part of that is because my thermo is rusty (never my best subject), but I also think part of the problem is presentation, particularly how the article relies heavily on certain formalisms without reviewing key concepts and equations. Often when I try to write about things after I've been deep into the calculations, I find myself throwing around a lot of details without getting the big picture across. I think this article has a little of the same problem.

On a related note, even though the equipartition theorem can be expressed more generally, I think it could help to start with the limited case of a quadratic hamiltonian and get the N/2*T result. Probably readers will be most interested in getting to that point, and one can always generalize the theorem later. It would be perfectly fine to say "for quadratic Hamiltonians, equipartition is the equal division of energy kT/2 into all possible modes." Gnixon 00:36, 14 April 2007 (UTC)

• Perhaps I am giving too strong an impression about Einstein. But he did propose E=hν both as the solution for the ultraviolet catastrophe (in 1905) and for the specific heat deviations from the Dulong-Petit law (1907), and his solutions were still not publicly accepted by any physicist in 1911, as described at the Photon article. Other physicists did not want to concede that light itself (or sound itself) had to be quantized; quite reasonably, they preferred the more conservative hypothesis that the emitters/absorbers were constrained somehow to emit energy in quanta.

After a little bit more reading, maybe I just didn't understand his role in pushing the photon concept. Interesting. Gnixon 00:36, 14 April 2007 (UTC)

When I have some time, I'll think more about your suggestions, and try to bring the article up to snuff. Thanks for your help! :) Willow 22:29, 13 April 2007 (UTC)

Glad to respond to other versions as the article evolves. Gnixon 00:36, 14 April 2007 (UTC)

P.S. Any suggestions for what the figures should look like?

Hmm, tough one. For one, maybe just take an arbitrary thermo picture for effect---it might be a little silly, but you could steal the animated picture on Physics. Other than that, maybe consider turning equations into images---you could do the general theorem and the ideal gas law, for example. You could show plots of the blackbody spectrum and perhaps something about the Dulong-Petit law. Those are all kind of random images, but at least they'd give us some eye candy. Gnixon 00:36, 14 April 2007 (UTC)

Re-review by Gnixon

Hi, Willow. I think the article has taken a huge step up since I last saw it. Nice work!

I'm so glad that you like it! :)

After a very quick re-reading, I have a couple comments:

• Great job presenting a simpler version of the theorem first, but I think it's been a little oversimplified. You've gone all the way to the ideal gas example where only the .5mv^2 energy is involved. I'd recommend instead stating the theorem as it applies to all quadratic forms of energy. It's not much harder to understand, and it explains applications like lattice vibrations (.5kx^2) and rotational contributions to gases' specific heats (.5Iw^2). I'd distinguish between the simplest form, (the theorem on quadratic energies) and the simplest application (the ideal gas).

Actually, the kinetic energy argument pertains to all atomic systems, not merely to the ideal gas. That's because the momentum and position are independent variables. I tried to clarify that and also to extrapolate as you suggest to other quadratic potentials. Please let me now if you like it!

Here's how the theorem was introduced in a book I have called "Introductory Statistical Mechanics" by Roger Bowley and Mariana Sanchez:

The equipartition theorem can be stated thus: every degree of freedom of a body which contributes a quadratic term of a coordinate or momentum to the total energy has an average energy of kT/2 and gives a contribution to the heat capacity of k/2.

I'd recommend saying it something like that. By the way, the text continued by briefly discussing the cases of monatomic and diatomic (two rotational, 1 vibrational DOF) gases, which might be useful here, too. Gnixon 14:29, 18 April 2007 (UTC)

I like the quote and I'll try to work it in, although it's not perfectly correct. The equipartition of kT/2 requires that the degree of freedom appears only as a quadratic term in the energy. For example, equipartition of kT/2 does not pertain to an anharmonic oscillator that contributes quadratic and higher-order terms to the energy, right? Willow 20:51, 18 April 2007 (UTC)

You're right, of course, about the anharmonic example. I think it's sort of a question of how you read the quote---certainly it can be said more clearly, and you seem to have done so. It'd be nice to phrase it without the "only" emphasizing the lack of other terms instead of the presence of the quadratic, but that's just nitpicking, and I haven't thought of a better phrasing.

More substantively, I like having the Maxwell-Boltzman distribution, but it's not exactly a derivation of equipartition. True, if you know the MB distribution, you can show the .5T stuff, but they both fundamentally come from the partition function, and moreover, you can show the .5T without getting the whole M-B distribution. What if instead we derived .5T using the full partition function formalism, but only analyzing a quadratic energy? (I'm thinking 1-d ideal gas, .5mv^2.) It would be obvious how it generalizes for more quadratic terms, and it would be simpler to follow than the fully general form that comes later (and it would prepare the reader for it). Sure, you'd have to use the partition function formalism, but with a link to the relevant article and some artful phrasing, that shouldn't be a problem.

By the way, it's truly a pleasure to watch this article evolve. Gnixon 02:07, 19 April 2007 (UTC)

Thanks so much, Gnixon! It's also a pleasure to work with you. :) I added something about the partition function for quadratic energies, but not right at the start; I was afraid of scaring off beginning students! Please let me know what you think of it. More generally, I'm thinking that references should be peppered throughout the article; what do you think? Thanks for your help as always, Willow 20:52, 19 April 2007 (UTC)

Thanks for adding that derivation for quadratic energies. I personally would have put it above the MB (which does have the advantage of being more concrete)---after all, who's going to be confused by a one-line derivation?---but it's not unreasonable to order things your way, and I understand the argument for it. I might fiddle with it just a bit so it mentions how the partition function formalism goes, but already it looks great.

References are always a good idea. Kittel is a standard advanced undergrad/grad thermo textbook. There were a couple others that saved my neck when I was studying for generals, but I can't remember them right this second. There's also that Bowley/Sanchez text. It's too wordy for my taste, but that could mean it has some quotable quotes, and the math isn't too hard. Gnixon 11:22, 20 April 2007 (UTC)

At some point it will probably help to trim the intro so it's more concise, but that's probably the last thing to do. Gnixon 11:22, 20 April 2007 (UTC)

• Going further with the simple/general divide, I'd recommend putting History immediately after the simple exposition, if possible. Many readers will be interested in the history, but might not get there if they're turned away by the general formulation. Likewise, you might consider trying to put issues of ergodicity and quantum failures above the more general formulation.

Good idea! Did this.

Again, great progress---it's showing promise to become a really fantastic article with your continued good work. I've got the article and this page on my watchlist. Do let me know if I can do anything else to help. Gnixon 14:58, 17 April 2007 (UTC)

Thanks, Gnixon! Looking forward to bringing it up to snuff, Willow 08:43, 18 April 2007 (UTC)

Mike Peel

Starting from the top:

• Introduction section
  • This has been written really nicely, so I don't really want to say this, but it needs references, and could ideally do with being shorter.

Leads do not need references, as they are summaries of information that is already cited in the article. --Cryptic C62 · Talk 22:40, 23 April 2007 (UTC)

Hmm; that's something I've not heard before. I've always tried to be in the habit of referencing everything, but so long as all of the information in the lead is discussed, and cited, elsewhere in the article, I guess that approach is OK. Mike Peel 23:58, 23 April 2007 (UTC)

• Systems with quadratic energies
  • It may be worth (briefly) explaining the reason for using the letter "H" earlier on in the article, where it first appears.

Explained H by "Hamiltonian". Willow 00:56, 24 April 2007 (UTC)

• • Also, explicitly state that $L=I\omega$ after it's used in equation ... grr; this is where I miss equation numbers. $H^{rot}={\frac {1}{2}}I\omega ^{2}={\frac {L^{2}}{2I}}$ (NB: you define it later, but it should be defined where it's first used)

Defined at first opportunity. Willow 00:56, 24 April 2007 (UTC)

• • $H^{pot}={\frac {1}{2}}kq^{2}$ where q is the deviation from equilibrium, such as the spring extension ... and k is ...? It's the spring constant here, rather than Boltzmann's constant, isn't it? (again, you define this several times later, but it should be defined where first used)

Replaced k/2 with A for consistency with following paragraph. Willow 00:56, 24 April 2007 (UTC)

• • More references in this section would be useful. Or does everything in this section come from McQuarrie (2000)? If so, which pages of that reference?

Gave page numbers; Callen reference is cited for general partition-function approach. Willow 13:25, 24 April 2007 (UTC)

• General formulation
  • Isn't this missing a factor of 1/2 in front of the $k_{B}T$? If not, why not?

No, the factor of two comes from the derivative of a quadratic Hamiltonian. As shown in the derivation sections, there's no factor of 1/2 needed.

It may be worth mentioning that in that section, or even better have a "see also" line or something pointing to the derivation for more information at the start rather than the end of the section. Mike Peel 23:58, 23 April 2007 (UTC)

Internal wiki-link to derivation? Willow 00:56, 24 April 2007 (UTC)

• History
  • Do references to the original papers by Waterson in 1843 and 1845 exist? The furthest back the reference goes at the moment is 1851.

Explained the 1845 reference; I'll need to dig around for the 1843 reference. Willow 01:17, 24 April 2007 (UTC)

Added 1843 Waterston reference Willow 16:31, 25 April 2007 (UTC)

• • "A few years later" - please be precise. How many years?

Fixed; sorry about that! Willow 01:17, 24 April 2007 (UTC)

• Anharmonic oscillators
  • Reference(s), please!

Two references added. Willow 16:31, 25 April 2007 (UTC)

• • I'm curious: is $m$ restricted to integer values, or is it any real number?

Any real number; it just comes taking the derivative.

How about with the Taylor series expansion? That seems to force m to be integer (and > 2).

Also, this section uses "m" for both the mass and for the index, which is a bit confusing. Mike Peel 23:58, 23 April 2007 (UTC)

You're so right; I tried to clarify this. Willow 00:56, 24 April 2007 (UTC)

• Non-ideal gases
  • "where the factor of two" - do you mean a factor of 1/2, not 2, here?

Exactly right!

• • "By an analogous argument" - is it possible to include a derivation of this, or alternatively link to somewhere else that does?

Included a reference; I could add the full derivation, but the article is already heavy on math, and it is just analogous to the prior derivation. I suspect that anyone who followed the ideal-gas derivation could do the non-ideal case themselves, especially given the answer and the analogous energy equation just above.

• Brownian motion
  • Does all of the content in this section come from a single reference?

The complete derivation is given by Pathria (1972).

• Stellar physics
  • Brief interlude: I can't read "assuming spherical symmetry" without thinking "assuming spherical cow..."
  • Back to work. Reference(s), please!
• Skipping over the derivations for now, as it's late and I'm tired. I'll have a look at them (and the rest of the article again) another day.

• Failure due to quantum effects
  • "which is required in the derivations of the equipartition theorem below" - I think you mean "above", not "below".

Oops, good catch!

In general:

• Please choose between using $k_{B}$ or just $k$ for Boltzmann's constant, rather than using both variably. This is especially important here, where you use k to also refer to the spring constant.

I thought that I had only used k_B for the Botzmann constant. Please point me to where I dropped the B subscript, thanks! :)

I've re-checked, and there were only a couple missing near the start. I've added the subscripts in now. Mike Peel 23:58, 23 April 2007 (UTC)

• Nice use of figures throughout. However, they're occasionally a bit big. Also, "Figure 1" is currently the second figure on the page, and "Figure 2" is the 9th. Is there a reason for this, or could all of the pictures/figures be numbered?

Figure numbering will get fixed once we settle on all the Figures; I'm thinking of adding a few more, and we may yet re-order the sections.

• More references are always good.

That's all for now. I'll try to leave more comments some other day, when I'm more awake. Mike Peel 22:27, 23 April 2007 (UTC)

Thanks muchly, Mike — you're awesome! Willow 23:02, 23 April 2007 (UTC)

A couple more (pretty minor) things (I was going to do these myself, but you seem to be doing a rewrite at the moment, so I'll avoid the edit conflicts):

• Equations should generally be punctuated as if they were part of the text, e.g. see [1].
• Use of (in German) and (in French) where appropriate in the references would be nice.

Thanks, Mike! Please go ahead; I'm about to go off to work! :) Willow 14:41, 29 April 2007 (UTC)

Mike Peel 14:36, 29 April 2007 (UTC)