Talk:Equipartition theorem

“Internal energy” or “kinetic energy of particle motion”?

What was here before said as follows in the first, defining paragraph:

“ The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system. ”

I added the underlining to “internal energy.” Note that internal energy includes all forms of heat energy, including the motion of free conduction electrons and the potential energy of phase changes. I'm not dead positive, but this can't be correct. Only the kinetic energy of particle motion is involved in the equipartion theorem.

Let's take the example of water (a molecule) melting. This form of potential energy (a quantum jump in a particular molecular bond from molecule to molecule) isn't shared amongst all the external and internal degrees of freedom. The equipartion theorem describes a straightforward concept; namely, how nitrogen (five total degrees of freedom) has five-thirds the specific heat capacity as do the monatomic gases such as helium (having only the three translational degrees of freedom) because the total kinetic energy of all the particle motions that the molecule currently has, is shared equally amongst all its currently available degrees of freedom. In other words, all of a molecule's degrees of freedom have he same temperature. However, I really doubt that the potential energy of phase changes has anything to do with this. Accordingly…

I changed “internal energy" to "kinetic energy of particle motion." Greg L 20:03, 31 January 2007 (UTC)

Greg, as far as I know, the equipartition applies to all degrees of freedom upon which the energy of the system depends quadratically. This includes potential energy (as long as the displacements from equilibrium are small). Therefore, the equipartition theorem is NOT simply limited to kinetic energy. In fact, when it is most commonly applied, (i.e. to vibrational modes in a molecule), you can plainly see this because the vibrational modes consum twice as much of the internal energy as the translational modes. So, I'm changing it back. Ed Sanville 09:33, 2 February 2007 (UTC)

Oh, and the reason it doesn't apply during phase changes is because the displacements are far enough from the equilibrium, that they become anharmonic and therefore non-quadratic. It's not because the theorem doesn't apply to potential energy components of the Hamiltonian. Ed Sanville 09:35, 2 February 2007 (UTC)

And, with regards to the nitrogen example you give... the classical equipartition theorem predicts a heat capacity of 7R/2, not 5R/2. The reason the experimental value is closer to 5R/2 is because the single vibrational mode, (which would classically contribute R to the heat capacity), has a very large energy spacing, (due to the high spring constant of the bond, and the low atomic mass of nitrogen). Therefore this vibrational mode behaves extremely quantum mechanically, and requires high temperatures to bring it into the classical regime. At high temperatures, the heat capacity of the N₂ gas is 7R/2. Also, the reason the electronic degrees of freedom never take part in the equipartition theorem is simply because they are never quadratic qith respect to the position or momenta of the electrons. Luckily, electronic excitations tend to have very wide spacings indeed... and are therefore always frozen out into the lowest energy level for the most part. This is why they tend to contribute little to the total heat capacity (for most molecules, anyway... Cl₂ and others are exceptions). Ed Sanville 09:45, 2 February 2007 (UTC)

Edsanville, I read your explanation but I don't see support for your argument that the potential energy of latent heat is included other than your statement that it simply is. I can find no source that supports what you're saying. Please see this link from the University of Manchester (4.8 The Equipartition Theorem). It says that the theorem holds “for vibrational, rotational and translational energies.” All it talks about is kinetic energy. It doesn't say one single thing about the potential energy of latent heat. You must cite a reputable source that supports what you're saying (potential energy is included). What you currently have makes no sense to me. Please note that I can find sites that say "internal energy," but they appear to be using the term more selectively than how the term is defined in Wikipedia; they all go on to address nothing more than kinetic energy. Greg L 07:25, 5 February 2007 (UTC)

Update: I've found this hyperphysics Web site and this one which give very succinct definitions of various thermodynamic terms. According to what I can find, internal energy includes all kinetic energy of particle motion within molecules, plus its "thermal energy" (translational kinetic energy), plus the potential energy of latent heat. However, absolutely nobody is saying that the equipartion theorem includes the potential energy of latent heat. Every single source I come across speaks only of the distribution of kinetic energy. Even this Equipartition theorem article goes on to discuss details only of kinetic energy and its distribution. The whole problem lies with the unfortunate use of "internal heat" (which too broadly encompasses too many forms of heat). If you want to say that the equipartition theorem includes not only kinetic energy, but also potential energy, you must cite authoritative references.

This should be very straightforward. The definition is simple:

“ The equipartition theorem states that for any bulk quantity (a statistically significant number of particles) of a molecular-based substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the degrees of freedom available to the particle. ”

Any definition that introduces the topic of "internal energy" improperly broadens this definition. Greg L 20:00, 5 February 2007 (UTC)

• Greg, my point was never that the equipartition theorem applies to latent heats of phase change. My point was that, since the equipartition theorem is only applicable to ideal gases, (and even then only at high temperatures), I don't think it's wrong to use the term internal energy in this context. But, kinetic energy is certainly not broad enough! The reason is that vibrational energies include both a kinetic and potential term, and they are treated as two degrees of freedom with respect to the equipartition theorem. So, I believe the best answer is a compromise between internal energy and kinetic energy, (neither of which is apparently 100% accurate here). Maybe we should move to the long-winded, but more accurate statement:
  

“ The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the translational, rotational, and vibrational partition functions of the degrees of freedom of a canonical ensemble of particles with a classical Hamiltonian composed only of terms that are quadratic with respect to the generalized coordinates and conjugate generalized momenta of the particles in the system, will tend toward their classical limits in the limit of high temperatures. ”

and then follow this with a simpler version:

“ In practice, this implies that the energy of a system of non-interacting molecules will generally allocate itself such that each translational and rotational degree of freedom recieves kT/2, and each vibrational degree of freedom recieves kT under the rigid rotor/harmonic oscillator approximation. ”

When you discuss latent heats of phase changes, I thought it was understood that you're talking about a realm way outside of the range of applicability of the equipartition theorem anyway. This is because in order to have latent heats of phase changing, etc., you have to have substantial intermolecular contributions to the Hamiltonian, which are pretty much never harmonic with respect to position or momentum.

In summary: The only requirements for the equipartition to be applicable is that the Hamiltonian energy of the system under consideration is quadratic with respect to some set of position/momentum degrees of freedom, and that the temperature is such that the system behaves relatively classically with respect to these degrees of freedom. These can involve both kinetic and potential energies, therefore the statement that the equipartition theorem only applies to kinetic energy is just as wrong, and I think more misleading, than describing it as allocating internal energy. Trying to apply the theorem to a system with phase changes would be very wrong solely because of these two requirements. I won't revert anything until I see your response. Ed Sanville 21:23, 5 February 2007 (UTC)

• Ed, Crikey!' What you propose seems like way, waaaay too advanced of language to use at this early point in the article. I can only offer you my advise. Editors should be mindful that Wikipedia policy (see WP:LEAD) is that articles on technical subjects — and in particular their lead sections — should be as generally accessible as possible for the subject matter. I think this article (all Wikipedia technical articles, really) would benefit from keeping the lead, defining paragraphs as simple as possible (plain-speak) so that someone like a high-schooler taking advanced science can actually get a little out of it before the article wades off into nine-syllable land. There would be no compromise to the article by simply reserving language such as what you propose for later in the article.

Also, I don't believe the equipartion theory applies to only ideal gases — or even gases in general. As far as I know, all molecular-based substances fall into equilibrium with all available degrees of freedom having the same kinetic energy (temperature). That’s kind of a “Well… duhhh” concept isn’t it? After all, if there was an available degree of freedom to, say a water molecule, and it didn’t have the same kinetic energy as the others, then by definition, it would be out of equilibrium. The Georgia State University’s Hyperphysics page titled Equipartition of Energy says nothing about gases; merely molecules.

“The reason is that vibrational energies include both a kinetic and potential term…”: Ed, I assume that you are talking about the potential energy of degrees of freedom that are still frozen out at a given temperature. That's why all the really good definitions speak as per what I proposed above: “the kinetic energy of particle motion is evenly distributed among all the degrees of freedom available to the particle.” What that means to me is that if a degree of freedom is still frozen out, then the total kinetic energy is divided into a smaller number of degrees of freedom. At some higher temperature, the specific heat capacity should diminish as latent degrees of freedom unfreeze: additional increments of kinetic energy would necessarily be divided amongst a greater number of degrees of freedom. I now realize that “degrees of freedom available to the particle” could be interpreted as meaning that the kinetic energy is divided amongst all degrees of freedom that could ever be available. You and I know that's not the case. So one might revise the sentence as follows:

“ The equipartition theorem states that for any bulk quantity (a statistically significant number of particles) of a molecular-based substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. ”

Simpler is better in lead sections.

All the definitions I can find on the Web that deal with the equipartion of energy keep to the following points: 1) it applies to molecules (there is no artificial limitation as to ideal gases); and 2) what is being equally divided is simply the total kinetic energy of translational, vibrational, and rotational particle motions (it would be improper to drag in the potential energy of latent heat); and 3) that kinetic energy is shared among whatever degrees of freedom are currently available to the molecule at a given temperature. The principal is really darn simple: all active degrees of freedom have the same temperature (provided that the standard set of caveats like “equilibrium,” etc. apply). I see no need to deviate from these points. Do you? Greg L 00:21, 6 February 2007 (UTC)

Greg, I still think the definition given in the first paragraph is too narrow. It leaves out the potential energy of vibrational interactions. I am talking about the potential energy of vibrational degrees of freedom that are not frozen out. Yes, the equipartition theorem predicts that the kinetic energy of a molecule is equally distributed among the classically accessible internal degrees of freedom, but it also predicts that the total energy of the molecule is equally divided among potential energy terms that are not frozen out and harmonic with respect to some internal coordinate. This has nothing to do with latent heats... that is just a red herring. Also, an ideal gas only implies that the voume of the molecule is negligible, and the intermolecular forces are negligible, both of which must be true in order for the equipartition theorem to be applicable. I hope I have explained myself clearly so far, but just in case, I would like to illustrate my point with a concrete quantitative example.

The equipartition theorem makes a quantitative prediction of the heat capacity of a diatomic ideal gas under the rigid rotor/harmonic oscillator approximation. The predicted per-molecule heat capacity is:

$C_{v}={\frac {3k}{2}} k k={\frac {7k}{2}}$

where k is Boltzmann's constant. The first term is of course from the three translational degrees. The second term is from the two (classically accessible) rotational degrees. The third term is ${\frac {k}{2}}$ for the kinetic energy of the vibrational mode, plus ${\frac {k}{2}}$ for the potential energy of the vibrational mode. With the current first paragraph, a person could miss this important point entirely.

For any case where the equipartition theorem is even remotely applicable, the energy that is partitioned is equivalent to the internal energy of the molecule. Limiting its applicability to kinetic energy is inaccurate. The equipartition theorem does not apply to liquids or solids at all. Phase transitions and latent heats are way out of the scope of the equipartition theorem.

Anyway Greg, I think the only reason we are debating this point is that you're coming at it from a thermodynamic perspective, and I'm coming at it from a statistical mechanics perspective. Being a thermo guy, you are taking offense to the original usage of the term internal energy because you think it erroneously broadens the applicability of the equipartition theorem. You would be correct... except for the fact that the article states that the equipartition theorem is only applicable to degrees of freedom which are both classically accessible (not frozen out), and quadratic with respect to either an internal coordinate or momentum. This rules out any system with any latent heat component to the internal energy automatically. Meanwhile, I am taking offense to your weakening of the equipartition theorem to only apply to kinetic energy terms in the energy. In any case, the equipartition theorem is only applicable to a very, very tiny set of systems, and does a terrible job of predicting the properties of most systems, (even simple diatomic gases like H₂), (see Heat capacity).

But, take a look at your own link for a moment... it discusses the fact that the equipartition theorem is applicable to the potential energy of vibrational interactions, (it should note that this is only true under the harmonic oscillator approximation... but it looks like an entry-level thermo website and we can forgive the simplification). I agree that my long-winded suggestion is far too complex for the poor high school students reading the article, but I really do think we should mention the potential energy aspect of the equipartition theorem, otherwise we are making it sound weaker than it really is. Ed Sanville 11:00, 6 February 2007 (UTC)

Good morning Ed. Well, I'm disappointed that there aren't more consistent definitions of the "Equipartion theorem." Here's some links: this one (#1) says the equipartition theory applies to only the three translational degrees of freedom (monatomic gases, as you’ve written before), …but this one (#2) says it applies to all degrees of freedom (molecules). So too does this one (#3), as well as this one from Wolfram Research — smart guys — (#4). And finally, the one I originally cited above (#5) (as you pointed out) says it applies to monatomic atoms. I had recently corresponded with a Ph.D. instructor at Gonzaga University here in Spokane. No, (if you clicked on the link), they’re not being politically incorrect, Spokane is that white!. The professor reviewed my The internal motions of molecules and specific heat paragraph and referenced the equipartion theorem in his comments. Clearly he thought the theorem applies to molecules. That’s why I recently added wording in the pagragraph mentioning the equipartition theory. That’s why I’m trying to make sure the two articles are consistent. I have no problem correcting the Thermodynamic temperature article, or this one. My objective is two-fold: make the articles consistent and correct.

Nowhere in the above-referenced links do I see any discussion of “the potential energy of vibrational interactions.” Please explain to a mechanical engineer-type mind what that means. If you look at what Ludwig Boltzmann discovered with regard to the equipartion theorem, he was simply saying that vibrational kinetic energy is distributed among all the available degrees of freedom. One sees this in his constant. He essentially discovered the underlying basis for the phenomenon of how different gases have different molar heat capacities. The simplest description I can think of to describe the phenomenon that the equipartition theorem addresses is as follows:

“ As heat is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a molecular-based substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. ”

There certainly seems to be no need whatsoever to expand the theorem mathematically to suggest that the potential energy of latent heat is somehow included in this. Nor do I see any basis for this notion in the here-cited links. Any concept of potential energy of any sort seems like it would fall under the rubric of internal energy. Greg L / (talk) 20:27, 6 February 2007 (UTC)

P.S. If you'd update your user information with your e-mail address, I could click on the “E-mail this user” link in the toolbox and send you a blind e-mail directly. If you replied from within Outlook, then we'd be able to exchange e-mails directly, bypass Wikipedia, and keep our e-mail addresses confidential. Greg L 20:29, 6 February 2007 (UTC)

As a preamble to my response, I have to stress that the equipartition theorem is a theorem. This means that it is a purely mathematical result that you can derive yourself from a model, (see Donald MacQuarrie's book on Stat Mech). The equipartition theorem will fail to give good results inasmuch as the model, (in this case a classical approximation of the internal energy of a system given as a sum of kinetic energies and harmonic potential energies), does not represent reality. Ed Sanville 20:40, 6 February 2007 (UTC)

Hi Greg. I just want to clear up a few things about what I have been saying, and what I haven't been saying. I never said the equipartition theorem only applies to monatomic gases, because it doesn't. What I said was that it can ony be reasonably applied to ideal gases, (not the same thing as monatomic)! In principle, you could apply the equipartition theorem blindly to almost any system, but it would be a rotten approximation to experiment. It works very well in the case of monatomic gases like the noble gases, of course. It also works reasonably well with some diatomics, provided that:

• the atoms are relatively heavy, (to avoid quantization and therefore "freezing out" of the mode)
• the spring constant of the vibrational mode is relatively small, (again to avoid quantization), relative to the temperature
• the temperature is low enough to avoid large displacements, (which would introduce anharmonicities in the potential energy of the vibrational mode)
• the temperature is low enough to avoid electronic excitations

It is difficult to find a diatomic gas with all of these characteristics, however. Sometimes you can assume the vibrational mode is "frozen out," and work with only the translational and rotational modes, giving an almost decent approximation. The reasons for deviations are all because the classical approximation is invalid. At low temperatures, the equipartition theorem even fails to describe the rotational modes because of quantization of angular momentum.

That is why most descriptions either stick with monatomic or diatomic gases, (or polyatomic gases with some low frequency harmonic vibrational modes). If you look up the derivation of the equipartition theorem in any statistical mechanics textbook, you will get a full treatment, which explains why there is an equal apportionment of internal energy among the quadratic degrees of freedom of a classical system. This was the math that Boltzmann originally worked out in his derivation. I would suggest reading the chapter in MacQuarrie about vibrational modes, and the harmonic oscillator approximation to get a good idea about why the equipartition theorem works for systems with low frequency harmonic modes, and why vibrational energy gets allocated to BOTH the kinetic and potential energy of the mode. I wish I had my copy still, but I sold it before I moved over here to the UK. You can find a simplified thermo version that mentions the potential energy of vibrational modes in almost any physical chemistry textbook as well. Anyway, I wrote most of this article originally, and I lifted the following text almost verbatim out of MacQuarrie's Stat Mech book:

In general, for any system with a classical Hamiltonian of the form:

$H=\sum _{i}^{m}{a_{i}p_{i}^{2}} \sum _{j}^{n}{b_{j}q_{j}^{2}} U(p_{m 1},p_{m 2},\dots ,p_{M},q_{n 1},q_{n 2},\dots q_{N})$

where $a_{i}$ and $b_{i}$ are constant with respect to all $q_{i<N}$ and $p_{i<M}$,

$q_{j}$ and $p_{i}$ are spatial coordinates and their conjugate momenta,

each degree of freedom $q_{i}$ and $p_{j}$ will contribute a total of ${\frac {1}{2}}k_{B}T$ to the system's total energy, resulting in a total of ${\frac {1}{2}}(m n)k_{B}T$ equipartition energy.

The equipartition theorem is valid only in the classical limit of an energy continuum. The equipartition theorem breaks down in the limit of large gaps between quantum energy levels, because it becomes more difficult to excite degrees of freedom which are highly quantized, such as electronic excitations in non-metals, vibrational modes with a large ratio of force constant to reduced mass, or rotational degrees of freedom about an axis with a low moment of inertia.

That explains exactly where the equipartition theorem applies, and where it doesn't. Notice how the second term is not a kinetic energy term. It is a quadratic in the variable $q_{j}$, which is a position coordinate term. This means that it is a harmonic potential energy term with respect to the internal coordinate $q_{j}$. This means that the equipartition theorem applies to this term as well as the kinetic energy terms. In fact, the only reason it applies to the kinetic energy terms is because they are also quadratic with respect to an internal coordinate (in this case the the momenta $p_{i}$).

Here is a sort of hand-waving derivation of the equipartition theorem using a harmonic oscillator, which clearly demonstrates how the equipartition theorem applies to the potential energy of classical harmonic vibrations: Derivation of the Equipartition Theorem

I had a long discussion with User:Sbharris about the equipartition theorem, and how to use it to predict the heat capacities of monatomic and diatomic gases, (as well as one can, anyway). Perhaps he can explain things better than I can. Ed Sanville 20:36, 6 February 2007 (UTC)

I think I've developed a "theory of mind" as to what the mathematics are doing (and you're thinking)! Let me try this out: If a molecular-based substance undergoes a phase transition, the resulting effect on total kinetic energy will be evenly distributed among the available degrees of freedom. Is that your position? If so, then I think it is still improper (incorrect) {or at least misleading} to say that potential energy is included. All phase changes do is change the kinetic energy available to be distributed. This much doesn't alter the fact that the equipartition theorem merely describes that whatever kinetic energy there is to distribute, is done so evenly across the available degrees of freedom. Attempts to introduce any notion of potential energy improperly intertwines the concept of internal energy into the discussion. Limiting the class of energy to simply kinetic energy of motion is analogous to saying this: “The net income will be evenly distributed among all the ball players who show up today.” Discussions of internal energy are analogous to this: “The net income that will be evenly distributed among all the ball players who show up today will be the gross income less expenses and taxes.” That’s what you seem to be saying. Correct me if I’m wrong. However, as originally worded, here’s what the analogy says: ”The gross income will be evenly distributed among all the ball players who show up today.” This, of course, is wrong. This link sums it up nicely in the terms I can understand. It says “…In thermal equilibrium, each microscopic degree of freedom has an amount ¹/₂K_BT of thermal energy associated with it.” Note that the term “thermal energy” is only the kinetic energy of motion. Greg L / (talk) 21:04, 6 February 2007 (UTC)

"If a molecular-based substance undergoes a phase transition, the resulting effect on total kinetic energy will be evenly distributed among the available degrees of freedom. Is that your position?"

No, that's not my position at all. My position is that if a substance undergoes a phase transition, it is way outside of the scope of the equipartition theorem. The reason, as I've said a few times before, is that phase transitions, and in fact any phase other than the gas phase involve large anharmonic contributions to the internal energy, as well as large quantum effects which completely destroy the model from which the equipartition theorem is derived. Ed Sanville 21:29, 6 February 2007 (UTC)

Clearly the problem is that I am totally unable to understand advanced math. I've actually developed two patented methods to calculate the equation of state of gases. However, this was via deep, deep, concentration into the subject. I also used a extensive use of a spreadsheet so I could get into the issue. I was really, really into the zone both times. I suspect that what was originally here was entirely correct. I suspect that what is here now is also entirely correct (just more focused). Perhaps, this article will benefit from having the targeted definition expanded upon in the article with what you're saying. I think that is what you first proposed a long time ago. Greg L 21:43, 6 February 2007 (UTC)

I think the main issue you seem to have is that you aren't realizing that the equipartition theorem is just a model. It's a model that doesn't work very well in 99.99999% of cases. It predicts one thing... and experiment gives a completely different value. Asking what the equipartition theorem would predict in the case of a phase change is a meaningless question, because the underlying classical model doesn't apply to condensed phases, never mind phase changes. But, the equipartition model deals 100% with the internal energy of a molecule. It deals well really well with translational degrees of freedom because they are never quantized, (well, as long as your container is big enough...). It deals moderately well with rotational degrees of freedom because they are finely quantized. It deals poorly with vibrational modes, because they are even more heavily quantized (usually). It fails completely with anything more complex, like van der Waals, hydrogen bonding, ionic bonding, and basically everything else in chemistry. This is because these cases are both anharmonic and highly quantized. But the fact remains that the equipartition theorem deals exclusively with the internal energy of a model system. Ed Sanville 21:51, 6 February 2007 (UTC)

Suggestions

The theorem as stated now in Wikipedia is incomplete because it is valid not only for "kinetic" energy, but for "total" energy.

One possible way of stating the correct theorem is:

"Each quadratic term in the Hamiltonian contributes with k_B T /2 to the total (kinetic potential) average energy in the classical limit".

The demonstration is quite easy: see for example page 43 of R. K. Pathria "Statistical Mechanics", Pergamon Press (Oxford 1972).

Note that wherever in the present Wikipedia artical says "kinetic energy" should be replaced by "total energy".

87.221.5.221 16:07, 15 February 2007 (UTC) Giancarlo Franzese (Professor of Statistical Mechanics at University of Barcelona) (Look for "Giancarlo Franzese" on Google to find more about me).

Thank you for your kind suggestions, Prof. Franzese. Hopefully, we improve the article and state the theorem in an even more general form. Please be patient with us; it may take a few weeks. Your suggestions would always be welcome. :) Willow 12:36, 24 March 2007 (UTC)

The moving model at the beginning, whilst useful, could be edited so that it can pause or something, as at the moment it is very distracting and makes it hard to read the beginning. — Preceding unsigned comment added by 137.222.122.2 (talk) 17:04, 9 December 2013 (UTC)

Comment: Needs much easier introductory material

Props to Willow for putting this up for review. But, bearing in mind that this is WP's single, only article on "equipartition of energy", it needs to start in much more general terms, give much more of an overview first, the most simple of examples, and a discussion in general terms of why equipartion fails, before it goes anywhere near the general case, hamiltonian coordinates, non-quadratic energies, etc. etc.

Remember, there are people coming to this page who may doing their first thermal physics course, and be meeting equipartition for the first time as a measure of temperature. That is the kind of level the article needs to start at.

At the moment it goes straight for a general statement. That is a mistake, and probably shouldn't even get into the first 1/3 of the article. Instead think of the article like a pyramid, starting at the top with just the shortest encapsulation of the idea, and then slowly building up more detailed presentation as the article goes on.

As a target, the opening WP:LEAD above the contents box shouldn't be more than about half a screen at most, should be a back of a postcard summary of the concept (with perhaps a mention of its flaws), and pitched in the simplest possible terms. Even below the contents, there should be a fair amount of introductory material and special cases and the way it can fail first, before going anywhere near anything like the level of maths and level of generalisation we're opening at currently. IMO. Jheald 21:32, 29 March 2007 (UTC)

Thanks, Jheald! :) You're totally right, and your detailed comments are really helpful for me. I'll try to simplify the article. The problem is that the principle "every degree of freedom gets ½k_BT energy on average" isn't always true. But perhaps we can get the gist of equipartition and its uses across without being too specifically quantitative.

I'd still appreciate a review of the science, too: did I leave anything out? did I put too much in? Is something incorrect? Thanks! :) Willow 21:38, 29 March 2007 (UTC)

I just want to say, I think you're making fantastic improvements to this article! It's still pitched at quite an uncompromising and challenging level, but with every edit you're making, it's going in the right direction. Lots of kudos to you for the work you're putting in here. Jheald 18:02, 2 April 2007 (UTC)

Yeay! :D Thanks so much — comments like yours make me blush, but also very, very happy. I'm off to track down some more references. Hey, if you have time, could you also review Encyclopædia Britannica on its FAC page? Thanks muchly! Willow 18:23, 2 April 2007 (UTC)

I agree entirely that this article is still too uncompromising and challenging, even though one of my recent edits did not help matters! Well, we have still a way to go, and Willow has been doing a great job... Geometry guy 16:47, 23 April 2007 (UTC)

How about a simple equation on a separate line in the introduction: U=1/2 * degrees of freedom * kT for one atom, or U=1/2 * degrees of freedom * RT for 1 mol? I came to this page trying to figure out whether equipartition theorem referred to this equation, to U=q w, or to something else entirely. Jojojlj (talk) 03:18, 20 October 2011 (UTC)

Waterston date

Impressive tracking down on the history!

The Waterston date looks like it should be earlier: either 1843 or 1845 or 1851, according to this webpage [1].

Maxwell and Kelvin's close friend Tait is said to have given "the first proof" of the theorem. (1886-1892). [But exactly what theorem?]

Some background on the state of the proposition in the late 1880s here (under "Kinetic theory of gases").

But still not quite clear what Tait was supposed by Kelvin to have proved. Jheald 07:39, 24 April 2007 (UTC)

Jheald 07:17, 3 April 2007 (UTC)

a style detail

The superscripts "kin", "grav", "pot", etc. should probably be in \mathrm{}. Michael Hardy 03:49, 29 April 2007 (UTC)

OK, I'll fix that; thanks for pointing it out! :) Willow 10:10, 29 April 2007 (UTC)

Simple question

As a statistician rather than a physicist, I was puzzled by the (non-Adobe!) p.d.f.s in Fig 2. Why is the y-axis labelled Probability density (s/m)? I expect the integral of a pdf (usually) to be unity. Is this a special notation? I suggest linking the phrase probability density function in any case.

Sorry: having read further I see that you do finally explain the point under "Derivation for kinetic energies". Maybe some explanation earlier on would be useful. --NigelG (or Ndsg) | Talk 10:25, 1 May 2007 (UTC)

Done. How does it look now? Geometry guy 11:01, 1 May 2007 (UTC)

That seems to clear it up! Thanks. Good luck with the FAC. --NigelG (or Ndsg) | Talk 18:42, 1 May 2007 (UTC)

Derivations

I'd really like to see this article centered around the "Derivation for quadratic energies" that is currently buried in the "Derivations" section. It's a simple 2-line derivation, and with a couple phrases about general use of the partition function, it could practically stand alone. Simply stating the quadratic form of the theorem with that derivation as the first topic in the article would give the reader plenty of context for reading the rest of the material here. "More generally," which that subsection currently opens with, would be more appropriate for the statement of the full theorem. Gnixon 19:21, 1 May 2007 (UTC)

I agree, but I think we might be alone at the FAC, where many want to lead with the history (which, in my view, is incomprehensible without a few definitions and examples)! Anyway, I've cut down the quadratic derivation to its bare essence. Could you fill in those few phrases to explain what the partition function is and how to use it? Then we might be able to prioritize this approach. Geometry guy 19:42, 1 May 2007 (UTC)

I can see the argument for starting with history, but as long as the theorem/derivation section stays concise, I prefer putting it first as per your argument. I'll take a shot at that section tonight or tomorrow if nobody beats me to it. Gnixon 19:51, 1 May 2007 (UTC)

Good luck. Meanwhile, I'll try to clarify the examples. I might also reintroduce the momentum point of view, but with more explanation. Geometry guy

Thanks. Re: momentum, I mostly just wanted it to stay consistent. Gnixon 21:17, 1 May 2007 (UTC)

By the way, the Maxwell-Boltzmann derivation isn't really a derivation at all, since the M-B distribution comes from the same place and takes more work to calculate. It's like solving the whole dynamics of a system to "derive" that momentum is conserved. Also, I'm not sure it's worth introducing the inverse temperature, beta---for our purposes, we can just as easily do everything in terms of the temperature itself. Gnixon 19:21, 1 May 2007 (UTC)

Yes I also noticed this when I tidied that section up. I just had a logical hat on and wanted to clarify how A follows from B, without worrying about whether A is harder to establish than B. I'd be happy to bin the section (the derivation involves computing a horrendous integral) and lead with the partition function derivation, although we probably should define the Maxwell-Boltzmann distribution somewhere, right? Geometry guy

What if we presented the M-B along with other ideal gas law stuff as a simple, explicit example of where the equipartition theorem applies? Gnixon 19:51, 1 May 2007 (UTC)

That would be great, since it would bring M-B closer to Figure 2. Geometry guy

As for inverse temperature, I agree we don't need it, but it is rather standard in partition functions, so maybe we should use this FA as a nice opportunity to introduce the concept. Geometry guy 19:42, 1 May 2007 (UTC)

If some section gets deep enough into partition functions that it's easier to write derivatives in beta, I think that would be a good place to introduce it. Remember we have to get the article to FA status first!  :) I continue to be amazed at how much content is here---Willow really has an eye for seeing all the connections. Gnixon 19:51, 1 May 2007 (UTC)

Yes, Wikipedia is lucky to have such a fantastic editor. As for the FA, I was pleased to see you haven't voted yet. Together with Willow, I think we can really nail this one, and then give our strongest support. Geometry guy 20:16, 1 May 2007 (UTC)

Blush — you guys are too nice. 3)

OK, this is my chance to say that I'm worried about introducing too many complications early on for math-phobic readers. Better to have few examples well explained for the lay-person than a fireworks display of mathematical morning glories, don't you agree? We don't want to dazzle our devoted readership, but to enlighten them, which I believe is best done gradually. Perhaps you'll allow other editors such as Awadewit, Ravedave or TimVickers to veto anything too obscure?

And most of all WillowW! I agree. I am against introducing complication early, and am working quite hard to make the article more comprehensible: in particular, I am trying to make absolutely sure that every single concept and variable is defined/explained before it is used; this is probably easier for a new editor to do than an established one. On the other hand, history is no substitute for careful explanations and examples, so I am trying to bring some (but not too many I hope) of the latter forward.

On the M-B distribution, I see it as a Boltzmann factor (normalized so that probabilities sum to one), so it surely can be considered a derivation on the same level as the partition function calculation, no? It doesn't pertain only to gases, but to all atomic matter, if I understand correctly. But I would be only too happy to learn better from you; I've gotten used to expecting fresh enlightenment and new perspectives from every round of edits! :) Willow 20:44, 1 May 2007 (UTC)

I have no idea, I'm just a humble mathematician :) Anyway, I think we should see what Gnixon does with the derivation for quadratic energies before deciding where to place it in the article. It might usefully go between the history and the general formulation, but the jury is still out. Geometry guy 21:09, 1 May 2007 (UTC)

Oh contraire, Willow! I'm learning from you, and you've at least once gently corrected a hasty and wrong statement by me. I appreciate your patience with my habit of stopping by infrequently to spout off my opinions. I agree it's an advantage that the M-B distribution follows directly from the Boltzmann factor without all the partition function formalism, but on the other hand, there are a number of steps involved in getting to f(v) that aren't obvious at a glance. The quadratic derivation seems simple to me because it's only using things I know---the definition of the partition function and the d[log(Z)]/d[beta] formula---but it's certainly true that it relies on formalism that may not be necessary. Of course, the progression of fundamentals goes (number of states)-->(extremize entropy)-->(boltzmann factor)-->(partition function), where the latter two may not be so ordered and are certainly already abstract. Hmm, sorry to cut myself off mid-thought, but must run. More later. Gnixon 22:06, 1 May 2007 (UTC)

I'm with Willow on this one. Start with prob ~ exp(-E/kT), put in a quadratic energy, normalise -> calculate equipartition. (Though it might be appropriate to put in these steps explicitly). Conceptually much less involved, and much more entry-level, than having to introduce all the machinery of the partition function (even if less neat). Plus more obvious to see how the assumption of the canonical distribution plays in. Jheald 23:33, 1 May 2007 (UTC)

I can definitely see that argument. However, there are two things that bother me about it:

• The most important fact---that .5T follows from E~v^2---isn't at all clear to me from the M-B derivation. For example, I can't do the last integral in my head, and I don't think it'd be enlightening to write it out. The partition function derivation I can follow in my head, and it's clear why we get the right result. I think neatness is an important issue for something like this.
• True, in the second case one must introduce both Z and dlnZ/dbeta, but is it that much worse than declaring that the Boltzmann factor gives the right probability distribution? The Boltzmann factor is equally abstract as Z, although I admit it has a more direct interpretation.

Maybe I'll try revising each section separately so either could stand as the first derivation (on some subpage to this talk page), since it's probably hard to judge until we get into the details. I notice I've done a lot of talking and not much writing, but of course, this part is easier. I'll try to do the harder part soon. More comments and suggestions are of course welcome. Gnixon 00:02, 2 May 2007 (UTC)

Recent restructuring

I didn't think it would be quite so hard to put the examples back in order, but I think I have done it now. I've moved the simplest quadratic energy examples to the first section. These are the examples that the reader needs to know to understand the history. The remaining examples have been combined in one section. I see no reason to call this section "Advanced" (that is just intimidating), but there is much that can be done to make it more friendly. I've done my best to maintain all (non-repetitive) material and links, but I may have made mistakes, since this work was not easy to do, so please check what I have done. I hope my edit summaries are helpful.

I'm done for today, but not in general: the history needs to be revised and the applications section needs to be shortened. Also I would like to be more bold with the general formulation and state it up-front, then derive its implications. The reverse approach just makes it seem more complicated than it is, in my opinion. Geometry guy 00:44, 2 May 2007 (UTC)

I think you did a great job. Maybe the revisions discussed in the above topic could be obviated by simply putting the shortest possible paragraph(s) before or within the current first section, where we would very simply state the general form of the theorem, followed immediately by its quadratic energies form, with only a brief reference to derivations below. Starting with literally only two equations, with one of them general and one in accessible form, couldn't possibly be confusing to the reader of an article about a theorem, right? We could even make a pretty box for them.  :) Gnixon 02:24, 2 May 2007 (UTC)

Thanks! I've now moved the molecular tumbling example into the first section as well, since it fits more naturally there, and can be treated more briefly. I've also reordered the general formulation as promised, and tried to clarify a couple of derivations. I'm not in favour of stating the general formula at the start of the article, because I think for the general reader the presence of an accessible equation does not (sadly) compensate for the appearance of an inaccessible one. Regarding the 'theorem' in the title, I think there is a case for moving the article to a more friendly name like "Equipartition" or "Law of equipartition" (after the FAC of course). For one thing, the article doesn't actually state a theorem with precise hypotheses; it rather uses phrases like "equipartition holds in situation x". A theorem always holds! On the other hand your final remark about a pretty box suggests a possible compromise: putting the formula in a fancy float with a punchy caption (in the spirit of a T-shirt with Maxwell's equations or "E= mc^2" written on it) if that is not too unencyclopedic! Geometry guy 15:45, 2 May 2007 (UTC)

Parenthetical remark to my main comment below: (I'm not sure using "theorem" is inherently problematic. Physicists frequently talk about theorems without formally stating the hypotheses, relying on context to imply them. E.g., the CPT theorem in field theory. My understanding is that the equipartition result was referred to as the same theorem under various derivations from different hypotheses. It may be an abuse of the language, but I think it's a common one.) Gnixon 16:54, 2 May 2007 (UTC)

A move to "Equipartition" (dropping "theorem") sounds like a good idea. If the article is about the theorem, I really think it needs to be stated in at least some form right up front (even in the lead instead of the first section). Gnixon 16:54, 2 May 2007 (UTC)

Agree on both counts. In particular, I'm well aware of the physical conception of a theorem, and it would be entirely wrong of me to impose a mathematical point of view! That was not my intention! Geometry guy 17:36, 2 May 2007 (UTC)

...although insisting on "form" for the element of surface area is a little excessive in this context.  ;-) Gnixon 18:26, 2 May 2007 (UTC)

Glad to see you share my view that a sense of humour is essential when editing WP! Anyway, my round of edits is essentially finished. I look forward to seeing what you can do with the derivations, but I'm ready to support. Geometry guy 18:38, 2 May 2007 (UTC)

Humor is clutch. Once again, I think your edits have made a big improvement---no joke on that point. I will try to fiddle with things once I get a reasonable block of time, but I think I could support now, too. I never would have thought an article on equipartition could be so interesting. Gnixon 22:34, 2 May 2007 (UTC)

Diatomic gas

This is another important part of the article: it plays a big role in the history section and shows up again in "Failure due to quantum effects". However, the article seems to say slightly different things in different places, and I am not sure what the actual facts are. Here are four quotes:

• "Idealized plot of the molar specific heat of a diatomic gas, which decreases from (7/2)R (the value predicted by equipartition) to (5/2)R and thence to (3/2)R, as its vibrational and rotational modes of motion are "frozen out"."
• "As described below, the theorem predicts that the specific heat of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mole·K). Experiments confirmed the former prediction, but found that the latter was instead 5 cal/(mole·K), which falls to 3 cal/(mole·K) at very low temperatures."
• "...the predicted molar-specific heat should be roughly 7 cal/(mole·K). However, the experimental value is roughly 5 cal/(mole·K) and falls to 3 cal/(mole·K) at very low temperatures."
• "Thus, the diatomic molecule has a molar specific heat of (5/2)R, instead of (7/2)R; the vibrational degree of freedom is frozen out at room temperature, and one rotational degree of freedom is frozen out because of symmetry. At even lower temperatures, the two remaining rotational modes are frozen out, giving a molar specific heat of (3/2)R,"

Now there are lots of diatomic gases in this world! Is it true that they all have molar specific heat 5 at room temperature? What happens at high temperature? The graph suggests that the msh goes up to 7. Have any experiments confirmed this? In any case, we should probably be a bit more careful how we phrase some of these assertions, but I don't have the expert knowledge to do that correctly. Anyone? Geometry guy 16:01, 2 May 2007 (UTC)

I don't see any factual errors, although perhaps it's unclear. Recall R~2 in calorie units, so 7/2*R=7 cal/mol/K. The freeze-out temperatures will vary for different molecules, right? (because the energy of the first excited state will depend on the molecule) Gnixon 16:35, 2 May 2007 (UTC)

Apologies: by working in calorie units, I obscured my main point. At present the article seems to state in places that the msh is never 7 (in calorie units, or 7/2 R in general), whereas in other places it suggests that at high enough temperatures it could be. As far as I know these "high enough temperatures" could be unachievable in the lab, or could be close to room temperature for some gases.

Anyway, I've tried to rephrase the history section so it does not make such emphatic claims, but please correct me if my rephrasing does not reflect the experimental history. I agree that the freeze out temperatures ought to be different for different molecules, but what are these temperatures, typically? Geometry guy 17:31, 2 May 2007 (UTC)

According to Baierlein's "Thermal Physics" text, typical diatomic gases at room temperature have the vibrational modes frozen out, but the others accessible, so the heat capacity is 5/2 in most cases. Gnixon 18:20, 2 May 2007 (UTC)

Hi all, I try to find a reference that gives the vibrational temperatures and/or the molar heat capacities. I'm pretty sure that the 7/2 R value will be achievable for all gases. Conceivably, they could fly apart (dissociate) at a lower temperature than the "unfreezing" happens, but I suspect that that won't happen. A typical covalent bond has roughly 80x more binding energy (~50 kcal/mol) than k_BT at room temperature (0.6 kcal/mol), and one should excite the vibrational modes long before the bond breaks.

The article is looking good! I might tweak a few things, but I'll wait a little longer. Symplectic manifold and differential forms, while no doubt beguiling for their fans, might be a little too daunting and "off-pathway" for our readers; what do you think? Maybe we could put symplectic manifold and other such topics in the See also's? Just a thought, Willow 21:29, 2 May 2007 (UTC)

Thanks, this is helpful. If any of my edits about diatomic gases are false claims, please correct them. Geometry guy 22:15, 2 May 2007 (UTC)

A reference for that stuff would be great---it might even provide a nice table for that section. I agree with tweaking away some of the more abstract language added by our friendly mathematician. Remember even most physics students have only had multivariate calculus and differential equations when they first learn stat mech. Also remember that different sections can assume different levels of sophistication---the general derivation might or might not be deserving of "manifold." Gnixon 22:29, 2 May 2007 (UTC)

Thank you Willow for looking up the data! This is nice and coherent now. Geometry guy 18:29, 3 May 2007 (UTC)

New math vs old

I'm happy for the two passing references to symplectic manifolds to be removed: the term "phase space" already covers this point, but I am not convinced it is any more friendly to our readers.

As for differential forms, the only occurence of this term is a casual remark I made in the edit history. In the text, the dot product is taken between a vector and an infinitesimal area without explaining what this means, so I added the word "form" to indicate that there might be something happening here! If anyone knows a better way to clarify this issue without entering into a large digression, please edit it! Geometry guy 22:15, 2 May 2007 (UTC)

Regarding forms, since it's a concept that may be unfamiliar to readers otherwise comfortable with that section, I'd prefer referring to a normal vector---it's closer to the elementary treatment in multivariate calculus. No need to wince if the application is simple enough that nobody will be confused! On the other hand, it may be awkward to try and insert a description of the normal, so maybe we could get away with glossing over the issue by just calling it the element of surface area with no further explanation---some will know what is meant while those who don't probably won't notice any problems. The "form" insertion is a nice way to handle it, too, but I'm worried that some student taking his first GR course will come along and insert a long parenthetical remark explaining what a form is in layman's terms---ugh. I could go with any of three solutions---no explanation, normal vector, or form---depending on details of how it looks. Gnixon 22:29, 2 May 2007 (UTC)

Sorry, the word "form" leapt out at me in the ideal-gas-law derivation; being one of my own nightmares, I thought that others would be scared of it, too. ;) But I'm flexible, too; I guess I would lean towards the finessed solution of "no explanation" to keep the focus, and just assume that Those who Know will know. Hunting for refs, Willow 22:45, 2 May 2007 (UTC)

Ah, so that is why you prefer to write "general formulation of the theorem" instead of "general form of theorem", Willow ;) Anyway, I've finessed the infinitesimal area using "element" as suggested by Gnixon. I've also removed symplectic manifold from the statement, though not the derivation. By and large, I've actually tried to reduce and clarify the math content in my edits, rather than elaborate it. There are surely fewer partial derivatives in the text than there were when I began! Geometry guy 11:47, 3 May 2007 (UTC)

I find the math vastly more readable since you've gone through it. My pledge to take a pass through things remains in effect, but it's been a busy week (sorry). Gnixon 17:28, 3 May 2007 (UTC)

Thanks: mostly it was just a case of replacing b=c=d=e=a by a=b=c=d=e or e=d=c=b=a (in an argument to show that a=e)! I'm glad you are still on the pledge: from my point of view, its mainly the derivation now that most needs some expert attention. Good luck! I'll probably register my support soon anyway. Geometry guy 18:29, 3 May 2007 (UTC)

Thank you very much, G-guy; the article reads much better with your touch! I have to wonder at myself sometimes, but I'm very glad that someone here can think straight and lay out roads through labyrinths. I'm a little embarrassed by the forms-thing, too, although I'm comforted by the fact that I'm not the only person in the world with math-phobia. ;) Usually I don't have to do anything harder than redesigning a dress for a different client or changing the gauge on a Fair Isle sweater pattern. Both of those can be hard enough! :) Willow 19:09, 3 May 2007 (UTC)

In my experience b=c=d=e=a is very common in the physics literature (and in a lot of maths too): it is meant to be read "b=c=d=e and everyone knows b=a, so why not just tag the equality on the end for short - after all, equality is an equivalence relation!" Which is true, but not so helpful to the neophyte. Ah well... Geometry guy 19:59, 3 May 2007 (UTC)

So no-one's offering $\mathbf {q} \wedge \mathbf {dS}$ where dS is the infinitessimal area bivector ? Much easier to visualise than forms!  :-) Jheald 20:30, 3 May 2007 (UTC) (-- more than happy to settle for area element!)

Calories

Can we lose the references to Calories? The definition of the Calorie is a nightmare, and AFAIK even chemists no longer use the unit, outside of North America. Please can we standardise on Joules instead -- it makes the energies so much easier to compare to other energies routinely quoted in Joules, particularly eg reaction activation energies. Jheald 23:13, 2 May 2007 (UTC)

Sure, if we're agreed. As you point out, Joules is the SI value and should be preferred. If I'm not mistaken, however, calories were used historically by the cited authors. Another (merely numerological) advantage is that using calories gives nice single-digit, nearly-integer values for the molar specific heat: 3, 5, 7, etc. that tie in neatly with the formulae. However, I suppose that 12.55≈13, 20.92≈21 and 29.29≈29 J/(mol K) are OK, too. Willow 09:33, 3 May 2007 (UTC)

Even though I'm a metric guy (for everything except the pint ;) and have already converted inches to centimetres at one point in the text, I have to admit that I like the use of cal/mole K for molar specific heats for exactly the reasons that Willow gives. Note that the calorie is only ever used for this unit. In fact there are almost no other explicit quantities in the whole article: just the beer haze (which uses Daltons, not kilograms for obvious reasons of size and context), a couple of facts about stars (one of which uses solar masses), and a quantum energy spacing (in electron volts, of course, not joules!).

For better or worse, this article is as much about the role equipartition played in the development of quantum mechanics as it is about the general theory: I think this is ample justification for using an historical unit. In my opinion this is for the better, since this historical aspect is one of the things that makes the article more interesting to a general reader. Such a reader is more likely to remember 3, 5 and 7 than 13, 21 and 29! Geometry guy 11:00, 3 May 2007 (UTC)

My personal plan is to skip metric and hold out for Planck units, but where professionalism demands pragmatism, I prefer SI. My first thought would be to stick with Joules (and give numbers with one digit after the decimal so readers don't think we're manipulating integers), but I can see an argument for calories. Gnixon 17:25, 3 May 2007 (UTC)

Quoting 12.6, 20.9 and 29.3 J/(mol K) doesn't seem unreasonable precision - it's not unreasonable to seek to measure a 20 K increase in temperature to 0.1 K precision. Wrt Geometry guy's point, what we want the reader to remember is 3/2, 5/2 and 7/2. The integer truncations of the numbers in J/(mol K) are unfortunate because the roundings obscure this ratio. But with one digit after the decimal, that's no longer a problem. Jheald 18:05, 3 May 2007 (UTC)

I quite like the fact that R ≈ 2 clears the denominator. Anyway, I agree that 3/2, 5/2 and 7/2 are the heart of the matter, so maybe we should actually try to give fewer formulae in explicit units. As I said already, these cal/mole K quantities are essentially the only explicit quantities in the text. In many places, they could be replaced by expressions like (3/2)R with no loss of information. Geometry guy 19:44, 3 May 2007 (UTC)

There's really no need to use units at all (except maybe one example if useful for getting a rough scale). The best thing would be to define the dimensionless specific heat (or whatever is the correct term) and show that experiments give, say, 2.48 /- .03 ~ 5/2 for a typical gas (or better, give a table, or best, give a plot of real data over temperature transitions). Do we have enough information available? Gnixon 23:11, 3 May 2007 (UTC)

Still work to be done!

Congratulations to all involved on the FA. It seems are work is not all done, however: the following comments by User:Awadewit are copied over from my talk page (with permission).

As a representative of the educated but not scientifically-trained masses (and an avid reader of popular science books), Willow asked me to look over the equipartition theorem article again for overly obscure language. As I wrote in my earlier peer review, I am not sure that this article is one that anyone will stumble on who doesn't have some mathematical and physical knowledge already (unlike, for example, natural selection), but I do believe that at least the lead of every article should make an attempt to be comprehensible by the non-specialist. I think that the lead for this article has improved, but, to me, the opening paragraph is overly specific (I am thinking here of for example, the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion). "Translation motion" and "rotational motion" may be obvious terms to physicists and mathematicians, but they were not to me (but perhaps that is just me). I would suggest that every attempt be made in the lead to explain equipartition in simple terms and leave the "meat" for the article. Unfortunately, I understood equipartition not from this article, but from my live-in physics expert who explained it to me after I read the article. There must be some way to convey the gist of equipartition to the non-specialist in this article - perhaps in a separate section? Awadewit ^Talk 20:04, 6 May 2007 (UTC)

Can we rise to the challenge to make the lead still less technical? Also, are we agreed to move the article to "Equipartition"? Then it remains to consider whether the derivations can also be clarified... Geometry guy 11:01, 7 May 2007 (UTC)

Equipartition and the rise of quantum theory

Third paragraph in the introduction "Equipartition's failure for electromagnetic radiation — also known as the ultraviolet catastrophe — led Albert Einstein to suggest that light itself was quantized into photons, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory." I believe it was Max Planck who suggested light was quantized not Einstein and it was not just due to the Ultraviolet catastrophe. See http://en.wikipedia.org/wiki/Ultraviolet_catastrophe —Preceding unsigned comment added by 65.32.59.88 (talk • contribs)

Thanks for writing! :) You're right about multiple factors being involved in Einstein's hypothesis, first expressed in his 1905 paper. However, Planck hypothesized that energy — or more strictly speaking, every change in energy involving electromagnetic radiation — was quantized, but not light. He thought that the law ΔE = hν was not a property of light, but rather reflected some restriction on the nature of the things that emit light or on the emission process. Planck opposed the quantization of light until roughly ten years after it was proposed, because he didn't want to have to change Maxwell's equations. Hoping that this clarifies it rather than mudddles it — thanks again for your letter! :) Willow 11:25, 16 June 2007 (UTC)

Strictly speaking Planck (by Willow's assertion) was right. Energy is unknown to within the infinite constant associated with the ground state energy which is totally unknown; the ground state energy arises from the 1/2 in the ladder description of quantum modes. And light and the Maxwell equations can be derived from a Lagrangian which allows only point interactions between particles at points on the light cone. (This is the Wheeler-Feynman universe.) In other words light is our fiction that we create to "explain" action at a distance. YouRang? (talk) 17:37, 18 December 2008 (UTC)

Weak or strong coupling between modes?

Dr. Benjamin Widom, in his "Statistical thermodynamics, A concise introduction for chemists" (Cambridge U.P.) writes: Weak interactions between the modes allows energy to flow between them, just as the weak interactions and rare collisions between molecules in an otherwise ideal gas are necessary in order to establish the equilibrium properties of the gas (p. 55). I believe this is common wisdom in the field: the fact that some degree of non-idealness has to enter in order the system equilibrates, even if it's later neglected. I am therefore puzzled by the theorem that states that coupling between normal modes must be strong enough. This seems to be at variance with the usual behaviour of crystals at low temperatures. --Daniel (talk) 07:38, 8 April 2008 (UTC)

Hi Daniel!

I'm dashing off to work, so I don't have time to really answer your question, since here's a brief try at an answer. If I understand it correctly, the KAM study showed pretty clearly that classical, weakly anharmonic systems are not ergodic unless you crank up the coupling of the modes. But the real world is not governed by classical physics, so those types of dynamical systems may not pertain at low temperatures? I'm not sure, but I would guess that the ergodic equilibration that occurs in low-temperature crystals occurs through quantum-mechanical Umklapp scattering of the phonons; but you should find a physicist to get a real answer. I'm curious, too, so I'll try to find out the next time I visit the library, or meet an obliging physicist. :) Willow (talk) 22:34, 8 April 2008 (UTC)

Motion

"The equipartition theorem can be used to derive the Brownian motion" it is impossible. Better "The equipartition theorem can be used to derive the mean square displacement of a Brownian particle" Alexander Mayorov (talk) 02:35, 20 October 2008 (UTC)

"metallic electrons" --> electrons in metal Alexander Mayorov (talk) 00:04, 25 October 2008 (UTC)

A grammatical tweek...

The last sentence in the third paragraph of the introduction seems a bit clunky:

"Along with other evidence, equipartition's failure to model black-body radiation—also known as the ultraviolet catastrophe—led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory."

In particular, "energy" is the subject of the clause, and is singular. The verb "were" seems to disagree, as it is not singular.

I suggest this:

"...led Max Planck to suggest that the energy of the oscillators in objects that emit light is quantized, a revolutionary..."

It has fewer commas, and seems a bit smoother. — Preceding unsigned comment added by 72.0.137.14 (talk) 14:31, 25 November 2011 (UTC)

Numbering images/figures for reference

The images in this article are manually numbered in their captions (e.g. "Figure 2. Probability density..."). The trouble with the manual system is obvious: if one figure is moved or removed then the whole numbering system goes bad. There's also the issue that some image numbers (e.g. the "Figure 1" in the header, which I just removed) go unreferenced in the article text, and serve only to clutter the caption. Is there not an automatic figure numbering system for images akin to the existing Template:NumBlk for equations? TSchwenn (talk) 23:51, 17 April 2012 (UTC)

Note that this feature was requested (and ignored) circa 2003. TSchwenn (talk) 00:03, 18 April 2012 (UTC)

See discussion at Wikipedia talk:WikiProject Physics#Numbered figures --TSchwenn (talk) 19:45, 29 May 2012 (UTC)

max plank motivation

In the intro it is suggested that max plank was trying to solve the ultraviolet catastrophe the page on this topic suggests otherwise hope this helps — Preceding unsigned comment added by 114.77.88.235 (talk) 12:38, 2 September 2012 (UTC)

Star formation

Such a collapse occurs ... when the gravitational potential energy exceeds twice the kinetic energy. I don't understand the factor of 2. Why is (3/2)NkT not enough? --egg 19:53, 4 September 2012 (UTC)

Kinetic or Potential?

My recent revision has been reversed here

The I made the original change to emphasise that it is only kinetic energy that is exhanged in the equipartition theorem.

The equipartion theorem is a vital part of equilibrium thermodynamics, equipartition and thermal equilibrium (uniform temperature) meet the common requiremnt for uniform distribution of thermal energy, however there is absolutely no corresponding requirement for uniform distribution of potential energy - one only has to think of chemical energy to realise this. Similarly a triple point cell relies on the non-uniform distribution of potential energy for its temperature stability.

I would like comments on this before revising the article (again!). --Damorbel (talk) 17:32, 5 April 2013 (UTC)

WP:PHYSICS review: A-level article

I'm beginning a sort of WP:Expert review process for articles independent of the featured article system which I've realized has problems. As such, I've rated this article a level 'A' which means it is of the quality that would be expected from a professional reference work on the subject. I say this as a person with graduate degrees in astrophysics, but I encourage others who have similar qualifications to make comments if they believe my judgement to be incorrect.

jps (talk) 02:13, 12 September 2013 (UTC)

possible FAR

Promoted 8 years ago, feels need a good review, mostly because there's a lot uncited text.--Jarodalien (talk) 14:49, 25 April 2015 (UTC)

How time flies! I'll be glad to add additional citations, if you will take the trouble to note here on this Talk page which sentences you would like a citation for. We can do it section by section if you wish. Don't be shy about making a laundry list; I was used to those from Adrianne. :)

Alas, I moved to Germany a few years ago and couldn't bring much of my personal physics library with me. Still, I may have books enough on hand; but I do ask for a bit of patience in finding English references. Willow (talk) 12:47, 14 May 2015 (UTC)

I think the language for the source isn't the issue here, only thing matter is reliable, thanks.--Jarodalien (talk) 15:48, 14 May 2015 (UTC)

Thank you! Could you please list the sentences that you feel require a citation? We can do it section-by-section. Willow (talk) 18:08, 18 May 2015 (UTC)

Sorry, I didn't notice this reply until today. "Specific heat capacity of solids" section, there's 3 paragraphs without footnotes; "Sedimentation of particles", first paragraph; "General formulation of the equipartition theorem", start at "The general formula is equivalent to the following two"; "Non-ideal gases", "Brownian motion", I understand this section are mostly explaining first formula, but there's too much sentence and explaining below, so I feel it's should be more clarify, which sentence come from where, this is only one example, I feel most section should done the same; also there's a same kind problem I found in lots of technical articles, feels like... textbook tone.--Jarodalien (talk) 03:49, 7 June 2015 (UTC)

Entropy expression in general proofs section

Can anyone explain why in the general proof section, $S=k_{B}\log \Sigma (E)$ instead of $S=k_{B}\log \Gamma (E,\Delta E)$? — Preceding unsigned comment added by Zasdfgbnm (talk • contribs) 05:17, 6 February 2016 (UTC)

Article issues and classification

The article is tagged as "needing additional references from April 2015", "unsourced statements from March 2018", and "disputed statements from May 2018". The B-class criteria #1 states; The article is suitably referenced, with inline citations. It has reliable sources, and any important or controversial material which is likely to be challenged is cited. Reassess to C-class. 17:11, 1 March 2023‎ User:Otr500

I believe that the article is actually reasonably well-sourced and the tags should have been removed some time ago. But a discussion about what needs sourcing, if anything, would be good to do. Qflib (talk) 20:36, 24 May 2024 (UTC)