Talk:Equipartition theorem

From Wikipedia, the free encyclopedia

A request has been made for this article to be scientifically peer reviewed in order to get expert opinions on how it may be improved, including verifying its content.

	Physics Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
B	This article has been rated as B-Class on the assessment scale.
High	This article is on a subject of High importance within physics.
Help with this template This article has been rated but has no comments. If appropriate, please review the article and leave comments here to identify the strengths and weaknesses of the article and what work it will need.

Contents 1 “Internal energy” or “kinetic energy of particle motion”? 2 Suggestions 3 Comment: Needs much easier introductory material 4 Waterston date

[edit] “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:

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 for the kinetic energy of the vibrational mode, plus 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:

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 to the system's total energy, resulting in a total of 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)

[edit] 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)

[edit] 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)

[edit] 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?]

Jheald 07:17, 3 April 2007 (UTC)