# Talk:Negative temperature

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## (sic)

The second sentence of the article goes as follows:

A system with a negative temperature is not colder than absolute zero, but rather it is, in a sense, hotter than infinite temperature (sic).

It is not clear what the "(sic)" at the end is supposed to mean, it doesn't seem to fit.

In written language, the abbreviation (sic) stands for the latin 'sicut'. It is used when for example a quote containing a spelling mistake is an exact copy of the original. By adding the warning '(sic)', the following signal is given to the reader: "Reader, this spelling mistake is not mine, this quote is copied to the letter."

Sometimes the signal '(sic)' is given when the quote is a statement that looks preposterous. Then '(sic)' signals to the reader: "Reader, it may be hard to beleive that this is a true quote, but it really is a verbatim quote.

Here, '(sic)' is used, but here the context does not involve quoting somebody else. Here the '(sic)' just seems to say: "Now isn't that a whopping surprise?"
--Cleon Teunissen | Talk 23:09, 25 July 2005 (UTC)

I've remove the (sic), because this is not being used in the context of a quote. If readers are going to find this implausible, this can be acknowledged in full prose and then elucidated until it is believable. -- Beland 07:09, 14 August 2005 (UTC)
This is a fine compromise. While the statement that there can be temperatures above infinity seems preposterous, the use of (sic) in the origional meaning of 'as it stands' has been diluted, and its use outside of quotations is rare to the point that it adds little information and would merely cause further confusion. Explicitly stating it is far better in every way. Nazlfrag 08:29, 9 November 2007 (UTC)
I don't remember ever hearing that sic could mean "as it stands." Sic generally means "thus," "so," or "in this way." I don't think it is a shortening of sicut (especially since sicut is simply a compound of sic and ut), but even if it is, sicut has a very similar meaning. Sic is (and always has been) inteded as a note by the editor that a given quote or reference is given as it was in the original. In fact, the usage to indicate that something given outside a quote is exact is a dilution of the original meaning. Either way, saying " . . . hotter than infinite temperature, as it stands." is hardly a sensible sentence, anyways. Eebster the Great (talk) 00:22, 29 March 2009 (UTC)

## Accuracy and clarity concerns

Ow...reading this article always makes my brain hurt. The section "Heat and molecular energy distribution" and the other sections seem to be discussing two different notions of negative temperature. The first distinguishes between different infinite temperatures, and the second is an artifact of accounting for different components of kinetic energy separately.

The first type of negative temperature is rather poorly explained, using apparently contradictory language. ("hotter than infinite"?) I'm worried that it's just plain incorrect. In any case, it needs to be explained a lot better.

It should also be explained that in no circumstances should any object or system have a negative kinetic energy, if indeed that's the case.

The cleanup-verify tag isn't exactly right, but you get the idea. I don't have the referenced book handy; if someone who does can check it and see whether or not it supports any of this article, that would be helpful. -- Beland 07:38, 14 August 2005 (UTC)

This isn't well-written, but the content seems correct. I don't have Kittel with me at home, but the context jives with what I remember. Salsb 23:04, August 15, 2005 (UTC)
The article is quite correct, but not very didactical. The "hotter than infinite temperature" remark is technically correct but not very useful. It would be equally correct, and perhaps more helpful, to say, "If a negative temperature system comes into contact with a positive-temperature system, heat will always flow from the negative temperature system to the positive temperature system. In this sense, a system at negative temperature is hotter than any sysetm at positive temperature." 130.89.204.173 18:40, 25 October 2005 (UTC)

This whole discussion needs to be much clearer. Since we are discussing the states of particles around a nucleus, the discussion needs to unambigously state that the notion of "negative temperature" is more realisically "energies lower than you can get just by cooling things down". If there were an article on "quantum thermodynamics", much of the discussion would go there.

Nothing that I see here is inherently wrong other than the "hotter than infinite". Remember that division by zero is not defined; it does not equal infinity. S Schaffter 22:08, 22 January 2006 (UTC)

Riemann sphere much? Melchoir 01:11, 11 February 2006 (UTC)
I also am bothered by the term "hotter than infinite". In what system? In the same system? Sure, depending on how you define terms. But that's just because there is a bound on the energy of the system (the higher energy, at some point, has a smaller number of states). So it may be best to say that negative temperature is hotter than infinite temperature "for that same system", or some such. - Paul Rimmer, 03 January 2008 —Preceding unsigned comment added by 128.146.37.103 (talk) 16:37, 3 January 2008 (UTC)

## Name of article

Thanks to GregGWood for making the redirect here to fulfill my requested article, negative absolute temperature. It hadn't occurred to me to search under this simpler name.

But I wonder if it's really a very good name. To physicists I suppose "negative temperature" is unambiguous, but to the layman, a temperature below the zero of whatever scale he's used to might be called a negative temperature. There's not even a mention of this linguistic possibility in the lead section. Not that I'm saying there should be (I can't think of a wording that's not just plain ugly), but if the article were at negative absolute temperature (with a redirect from negative temperature) it should head off such a possible confusion. --Trovatore 23:00, 16 March 2006 (UTC)

Hmm... I think "Negative temperature" is okay. If you Google ("negative temperature" -coefficient), I'd estimate only about 10% of hits are talking about subzero temperatures on some scale, so there doesn't seem to be a discrepancy with common usage. Conversely, that search gets 245,000 hits, while inserting "absolute" gets just 252, so "negative temperature" really is the more common name for the concept.
As long as you can do it without using the words "this article" or similar "metalanguage". I really dislike that. If an article describes the content well enough, it shouldn't have to describe itself. --Trovatore 23:46, 16 March 2006 (UTC)
Indeed. Let's see... Melchoir 23:58, 16 March 2006 (UTC)
Nice job! --Trovatore 00:35, 17 March 2006 (UTC)

## new comment

I'm reverting this article, to see if it has been vandlised--WngLdr34 23:40, 18 November 2006 (UTC)

Sorry, WngLdr34, but I can't make out what you mean. You don't seem to have edited the article. By the way, new comments on talk pages should go to the bottom of the page. Welcome to WP! --Trovatore 22:28, 19 November 2006 (UTC)
To properly check for vandalism, you can examine the difference between any two edits on the history tab without reverting anything. Thanks for looking out for vandals though! Nazlfrag 08:36, 9 November 2007 (UTC)

I think I know what he means: Negative Temperature doesn´t exists. Temperature is only defined in equilibrium states and all the examples just look like Negative Temperature if you just watch a small aspect of the system whitch is not in equilibrium with the rest of the system.

You could try the same complaint about positive temperatures, especially in the theory of heat flow, which relies upon the concept of temperature gradients. A small system does not have to be in equilibrium with "the rest of the system" -- its surroundings -- to have a meaningful temperature. Its internal relaxation time must simply be shorter than its relaxation time due to external heat transfer. This physical, quantitative criterion has been experimentally verified in many nuclear spin systems. They really are sufficiently isolated from their lattices to have their own temperatures, and sometimes these are negative.
Granted, the article doesn't explain all this, but it will when I work up the energy. Melchoir 18:48, 29 November 2006 (UTC)

Unless i am mistaken, the text refers to page number 462 of 'Thermal Physics'. I quote from paragraph "Heat and molecular energy distribution":

As Kittel and Kroemer (p.462) put it, "The temperature scale from cold to hot runs +0 K, . . . , +300 K, . . . , +∞ K, −∞ K, . . . , −300 K, . . . , −0 K."

Looks like it is referring to this (non-public, but commercial) book: (first one mentioned in the references)

Note: I am not experienced regarding contributing to wikipedia, please excuse me if i did something wrong. I will not take action and remove that part of the text myself, please doublecheck my findings.

83.98.234.114 09:50, 14 December 2006 (UTC)

You're allowed (even encouraged) to quote from reference materials, even if they're copyright. This has nothing to do with Wikipedia per se; it's a general rule. The borderline where citation becomes infringement is fuzzy and ill-defined, but one brief sentence out of a whole book isn't even on the radar screen. See fair use for more info. --Trovatore 17:41, 14 December 2006 (UTC)

## new reference

you all should take a look at this site

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/neg_temperature.html

it has more stuff about negative temperature, although i don't know how to add references, all i do on wikipedia is suggest things and edit grammar ;) —The preceding unsigned comment was added by 68.188.54.173 (talk) 04:11, 18 February 2007 (UTC).

The link appears broken, I assume this is the correct link: http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/negativeTemperature.html

It contains the Usenet physics FAQ entry for negative temperature and is a great introduction. It would make a useful reference and caters to laymen. Nazlfrag 08:41, 9 November 2007 (UTC)

FWIW, As of 02 Aug 2016, the former link (by 68.188.54.173) works while the latter link (by Nazlfrag?) does not work. JimScott (talk) 19:30, 2 August 2016 (UTC)

## Ah, the circularity of numbers...

I liken these superinfinite negative temperatures to such identities as 1 + 2 + 4 + 8 + 16 + 32... = −1 (see Geometric series: a = 1, r = 2, sum = 1/(1 − 2) = −1)

212.137.63.86 (talk) 11:17, 21 August 2008 (UTC)

(... though if you read the article you cite, you will see that the formula is not valid for your example, so your "identity" is a falsehood!) Dbfirs 11:28, 21 August 2008 (UTC)
To be fair, there are valid interpretations at 1 + 2 + 4 + 8 + · · ·, and Divergent geometric series is a more relevant general article to cite. Also note that Euler himself made exactly the same point as 212.137.63.86; see [1]. Melchoir (talk) 19:26, 21 August 2008 (UTC)

## Upper bound

The articles reads "Since there is no upper bound on momentum of an atom there is no upper bound", however a concept of absolute hot does exist... XApple (talk) 17:40, 13 June 2009 (UTC)

## What is going on with enthropy of an isolated system, in which one element has negative temperature?

Take the energy adder (radio frequency techniques) and the nuclear spins system. The energy adder gives energy to the spins system. Enthropy of spins system decreases; therefore, enthropy of the energy adder increases (whilst the energy adder is outputting energy). Does it mean that this energy adder also has negative temperature? Or the system isn't isolated enough, and enthropy of the energy adder decreases while enthropy somewhere else increases?

It's just confusing: numbers larger than infinity are negative. One more confusing example:

fraction ${\displaystyle {\frac {1}{x}}}$, where x is larger than positive infinity, is smaller, than fraction${\displaystyle {\frac {1}{\infty }}=0}$.

Wikiwide (talk) 11:08, 20 November 2009 (UTC)

## Laser is not a system with negative temperature

Temperature is defined for systems in thermal equilibrium. A laser with population inversion created and/or maintained by pumping is a system out of equilibrium and the notion of temperature does not apply. The description of population inversion by a Boltzmann factor with negative T is a mere formal similarity and it has nothing to do with negative temperature. Lesche (talk) 21:38, 13 July 2010 (UTC)

Well, there's equilibrium and equilibrium. If you insist on perfect equilibrium, then you can't define any temperatures until the heat death of the universe. My understanding is that the subsystem of excited-versus-unexcited atoms in a laser has its own equilibrium, which is not of course in equilibrium with the translational/vibrational/rotational modes of those atoms, and that that subsystem is indeed at negative temperature. (But I'm not an expert on the topic.) --Trovatore (talk) 21:42, 13 July 2010 (UTC)

http://www.sciencemag.org/content/339/6115/52 — Preceding unsigned comment added by 84.183.118.41 (talk) 13:33, 4 January 2013 (UTC)

## If this isn't a crock of hooey, why doesn't it have any citations?

This article has been tagged for its complete dearth of citations for nearly four years. It is clear that this is because the concepts within are utterly and complete bullsh!t, but that no one has been brave enough to call it what it is. I'm going to submit this to the Wikipedia authorities for deletion if this situation isn't fixed by the fourth anniversary of this tag. 50.193.171.69 (talk) 02:29, 7 January 2013 (UTC)

The information is not unreferenced. It's true that it doesn't have inline citations, which is arguably a flaw, but if you'll trouble yourself to find the references you'll see that it is all true. (Inline citations are not any better than general references, if you don't bother to look them up.) --Trovatore (talk) 02:34, 7 January 2013 (UTC)
The problem with a lack of inline citations, though, is that we don't know which information in the article applies to which reference or whether the entire article is properly referenced at all. If one wanted to look up more about a specific part of the article in the references, they wouldn't know which one to look at. Someone who is familiar with or has a copy of the references listed really needs to turn them into inline citations. SilverserenC 11:37, 7 January 2013 (UTC)
That's fine and useful. My point stands, though; if people want to write nonsense and give references that don't actually support it, as the IP was suggesting, they can do that just as well with inline cites, if no one actually bothers to look them up anyway. A cursory attempt to do, with the references that already exist in the article, so would convince any reader, not necessarily of correctness, but at least that the article wasn't making stuff up. --Trovatore (talk) 11:50, 7 January 2013 (UTC)
And that's a fair point, Trovatore, because I came across this by hitting "Random Article" (which is how I usually come to articles) and what I read seemed more counterintuitive than quantum physics, which itself can take some getting used to. I still don't understand the article, but I realize that my reaction was formed when I read what appeared to be nonsense combined with a four-year old tag asking for citations. Anyway, I'm glad we all understand now. 50.193.171.69 (talk) 16:45, 7 January 2013 (UTC)

I added two citations to "The Laws of Thermodynamics: A Very Short Introduction by Peter Atkins, Oxford University Press, 2010" which is a very nice and accessible book that discusses this. (Pages 89-95 are on negative temperature with a reference to pages 10-14 in chapter 1 on Boltzmann's distribution where it discusses that beta is the reciprical of temperature.) RJFJR (talk) 17:10, 8 January 2013 (UTC)

Pretty much every sentence expresses an idea in one of these three of the listed sources. Kittel and Kroemer, Atkins, and Ramsey, Norman. It is simply a matter of labeling169.232.131.133 (talk) 23:18, 9 January 2013 (UTC)

## Magnetars

"Noninteracting two–level particles Entropy, thermodynamic beta, and temperature as a function of the energy for a system of N noninteracting two–level particles. The simplest example, albeit a rather nonphysical one, is to consider a system of N particles, each of which can take an energy of either +ε or -ε but are otherwise noninteracting."

It is possible to have physical example, if we have N fermions with spin 1/2 and in a huge external field B (Kinetic energy of the particle<< Magnetic energy) then we have the example. kinetic energy of electron is 3/2KT=3/2*1.38^-23T J K−1 Magnetic energy with intrinsic magnetic moment m is m.B=-928.476377 × 10−26 B J·T− So with B=10^8 Tesla (Magnetars are characterized by their extremely powerful magnetic fields, which can reach the order of ten gigateslas.) and T=300 K we have kinetic energy of electron =6.21 ^-21 J Magnetic energy = 9.2847^-11 >> kinetic energy 178.128.142.252 (talk) 22:41, 13 January 2013 (UTC)

I'm a little unclear on why you have posted this here. Does it relate to something you feel should be changed about the article? If not, this is probably not the place to post it. --Trovatore (talk) 22:57, 13 January 2013 (UTC)
Oh wait — the text you started with is existing article text, and you're objecting to the word "nonphysical"? Sure, we can take that out. I don't know why anyone would claim it's nonphysical, magnetars or no. --Trovatore (talk) 23:03, 13 January 2013 (UTC)
Actually, now that I look at it, I think the point of the "nonphysical" was that the particles are noninteracting. Physical particles, I suppose, are going to interact. Is there some reason they wouldn't, near a magnetar? --Trovatore (talk) 23:05, 13 January 2013 (UTC)

## New Article

"Science News" February 9, 2013, Page 10. "Hottest temp is as negative one: New record reached with ultracold gas at high temperature" by Andrew Grant. (It's in the "In the News section".) RJFJR (talk) 20:06, 11 February 2013 (UTC)

found the online link RJFJR (talk) 20:07, 11 February 2013 (UTC)

## Negative temperature depends on how entropy is defined

According to this article, apparently there are two ways to define temperature, and negative temperatures only come out in one version. (These two temperatures are an old concept, dating all the way back to Gibbs). Perhaps this is relevant to this article. The authors of that arxiv paper also claim that the thermodynamics which admits negative temperatures is incomplete somehow ... I'm not sure I'm convinced about that, though. --Nanite (talk) 16:37, 27 August 2013 (UTC)

I've added some disclaimers in the article about the validity of negative temperature in the meanwhile. Hopefully an expert can sort things out

thedoctar (talk) 08:22, 13 June 2014 (UTC)

A recent very readable paper by Frenkel and Warren (http://arxiv.org/abs/1403.4299) shows the concept of negative temperature is on very solid footing, and that the defnition of entropy employed by Dunkel and Hilbert has some very undesirable properties.--Allard (talk) 16:47, 23 October 2014 (UTC)

• Well, let's wait till the reviewers evaluate this paper, it's interesting to see that the paper is "conditionally accepted Am J Phys", and I'm sure that Dunkel and Hilbert will comment on it. --Fedor Babkin (talk) 08:37, 25 October 2014 (UTC)

Another very readable recent new article by Swendsen and Yang (http://arxiv.org/abs/1410.4619) (building on some arguments by Frenkel and Warren but also providing new arguments) shows that the 'inconsistencies' brought forward by Dunkel and Hilbert (DH) are all 1/N and therefore swamped by thermal noise and vanish in the thermodynamic limit and are for sure immeasurable for all the systems sizes so far realised in experiments. They also show that DH volume entropy is inconsistenst with thermodynamics. Becbuilder (talk) 09:25, 5 November 2014 (UTC)

I think the clarification does more harm than good. I teach Statistical Mechanics, and while the paper by Dunkel and Hilbert raises some interesting points, it does not represent the textbook view of entropy and temperature. Their definition of the "Gibbs entropy" is at odds with the definition of Gibbs entropy that is cited in Wikipedia, for instance. Furthermore, their version of the Gibbs entropy is problematic because it leads to violations of the 2nd law of thermodynamics, whereas the Boltzmann entropy does not. While the result is worth mentioning in this article, I think the definition of negative temperatures that comes out of the Boltzmann definition of entropy is on solid footing. In the thermodynamic limit (i.e. systems composed of many particles) the Boltzmann temperature and Boltzmann entropy are perfectly consistent with the laws of thermodynamics. Diary of Finknottle (talk) 09:33, 31 August 2015 (UTC)

Likewise here. From a statistical-inference perspective, there is no reason that the Lagrange-multiplier (reciprocal-temperature) for an observable (in this case energy) can't be negative, and the spin-system problems to which it is applied are cases in point. Frenkel & Warren is therefore on more solid ground, with Dunkel & Hilbert in good style but unwittingly prolonging the dissonance, although I take issue with stuff in both papers, including their attempt to pull historical authors (Boltzmann and Gibbs) into their dispute since those authors had very different contexts in which to work. Some of the culpability there goes to Dunkel, Hilbert, and the Sokolov "news & views", which reminds me of Nature magazine's past propensity for sensationalism. On the bright side, it probably helped inspire a spirited response in AJP. Thermochap (talk) 19:50, 6 October 2015 (UTC)
To be more specific, the papers are about use (for given observed average or expected energy E) of the number of states within an energy interval Δε about E (Frenkel & Warren like this) versus the number of states less than E (Dunkel & Hilbert like this). All agree that for many applications the distinction is immaterial. In general it depends on what your observations constrain, although if you really do know that E is in the energy interval Δε about E, then the multiplicity of states at energy above or below that range may not matter. This assertion doesn't just apply if E is energy, or even require that E be conserved, but that is separate story. Thermochap (talk) 13:14, 7 October 2015 (UTC)

Two and a half years on from my original comment, I believe there is a clear answer to the question: it is a false dilemma, as neither the "volume" nor "surface" temperatures are correct! There is actually a third definition of temperature, used in the canonical ensemble, which is unambiguous and unanimous, and actually functions like temperature should. (And it can be negative in energy-bounded systems (lasers, spins, etc.).)

The problem with both the "volume" and "surface" temperature definitions is they presume the microcanonical ensemble where it is energy that is fixed instead of temperature, and there is no quantity which really holds up to the qualities we expect of temperature (such as the zeroth law of thermodynamics). This is especially true in systems with few degrees of freedom; the only reason we think they are temperatures is that they often converge to a temperature-like quantity, for many degrees of freedom. A while ago I added some 113-year old content to the microcanonical ensemble article about this (from Gibbs' book that coined the term "statistical mechanics"). Ironically, Gibbs was aware of both "surface temperature" and "volume temperature" definitions and spelled out exactly the flaws in both of them. --Nanite (talk) 21:56, 7 December 2015 (UTC)

I am an expert and I kindly ask the participants of this discussion to do two things:

1. Please note that the microcanonical ensemble is the starting point of all statistical mechanics. All other ensembles derive from it. The microcanonical ensemble is therefore not a "problem", not even experimentally. In fact isolation from the environment is the main advantage of working with ultracold atoms because they allow to realize systems that evolve to a very good approximation unitarily in time.

2. Please read the Nature Physics paper by Dunkel and Hilbert, linked above until Equation (7). It is based exclusively on rigorous mathematical derivations starting from the microcanonical ensemble. From then on it's practically just taking partial derivatives. What they show first is that if you start with the volume entropy in statistical mechanics instead of the surface (or Boltzmann) entropy you can connect statistical mechanics to thermodynamics without any complications. If you start from the Boltzmann entropy this is not the case. Equation (7) connects thermodynamics and statistical mechanics. The surface entropy does not satisfy Eq. (7), whereas the volume entropy does. This is the shortest rigorous way to see that the Boltzmann/surface entropy cannot be the fundamentally correct entropy.

All arguments put forward by Frenkel and Warren, Swendsen and Yang, Schneider et al. and some others are faulty and have been disproven in detailed responses. I give some of the references to the publicly accessible preprints. I hope there won't be many more necessary.

http://arxiv.org/abs/1507.05713

http://arxiv.org/abs/1408.5392

http://arxiv.org/abs/1408.5382

http://arxiv.org/abs/1403.6058

Some of the discussion on this page is just rephrasing arguments of Frenkel and Warren, personal "opinions" and badly researched. For example, the volume entropy is the only known definition of entropy that is consistent with the laws of thermodynamics, especially also the 2nd, in contrast to what User Diary of Finknottle quoted. See the third linked paper http://arxiv.org/abs/1408.5382. The Boltzmann entropy violates the second law, as shown in the same work. This might be good reading for the User Thermochap. In contrast to the arguments by the proponents of the Boltzmann entropy which resort to prosaic arguments, the works I linked to above are based on rigorous but very simple mathematical derivations. If you are interested please take the time and read them and then improve Wikipedia.

Currently the Wikipedia article is complete rubbish. There is no false dilemma here either, User: Nanite. The Boltzmann entropy is not compatible with thermodynamics, but more often than not we don't notice the difference, between the Gibbs and the Boltzmann entropy. A side effect of consistency with thermodynamics is that there cannot be a negative temperature. It's time to understand that some of the material presented in textbooks for a few decades has to be revised. But I don't want to participate in any editing wars. Last but not least: I am not an author of any of the above papers. Davidwuelfert (talk) 21:40, 3 March 2016 (UTC)

I understand where you're coming from. You think the canonical ensemble can only be derived from the microcanonical ensemble as the "mother of all ensembles", to quote the arXiv:1507.05713 paper. This is an unfortunately common viewpoint put forward by numerous modern undergrad-level statistical mechanics textbooks, and is closely linked with the viewpoint that a canonical ensemble is defined by a system in continuous physical contact with a large reservoir. As if the reservoir is somehow necessary to obtain this probability distribution.
There is a related and illogical view that any isolated system (not in contact with a reservoir) must therefore be in the microcanonical ensemble. But a moment's thought shows why this is false: any system that was previously a canonical ensemble continues to be in a canonical ensemble even after it is isolated. Its energy does not suddenly and spontaneously become exactly known to us, unless we measure it. The notion of reservoir contact is really only a helpful imaginary tool, which gives an easy image to students of the meaning of temperature and chemical potential.
The above problems I mention are due to mutations of the original, simple viewpoint of Gibbs when he defined the canonical ensemble in his 1902 Elementary Principles book. The canonical ensemble — a probability distribution — was invented to follow the property that when two systems with the same β parameter (temperature) are brought into weak contact and allowed to exchange energy, their probability distributions do not change. This is the statistical notion of equilibrium between two systems. It does not matter if one or either of the systems is macroscopic nor a reservoir. Nor does it require that the contact is made, only that it could be made and that it would not matter. Here Gibbs defines temperature in an elegant way that applies across classical and quantum mechanics, from macro to micro.
Note that two microcanonical ensembles at the same "temperature"—by ANY definition—are not in any way in equilibrium with each other and cannot even manage to follow the zeroth law of thermodynamics. I would ask: how can you call a quantity "temperature" or "entropy" simply because it obeys a first law of thermodynamics? That's a very short-sighted viewpoint. You cannot even build a heat engine with a microcanonical ensemble, and so what physical application do these mathematical variables even have?
Note also that it is not necessary to invent the microcanonical ensemble before the canonical ensemble: in Gibbs' book the former comes in chapter 10, and the latter was already defined in chapter 4! And again, I stress that Gibbs has preempted all of these silly debates about microcanonical temperatures in his chapter 14. People just do not read the old literature and hence they keep repeating the same debates over and over.
Now, to be honest, many experiments with negative temperature involve isolated systems which are neither described by a microncanonical ensemble (they would have to have EXACTLY known energy, and this is never possible in any system, ever), nor are they described by a canonical ensemble (they would have to follow EXACTLY a canonical probability distribution). Really, they are short-lived transient states with no exact description in terms of any of the familiar textbook ensembles. Yet, if someone demonstrates that they can prepare a system that can be described with high accuracy by a canonical ensemble of negative β, wouldn't you say this system pretty much has a negative temperature? That is something I would accept, pragmatically. As to whether this has been accomplished yet, I am not familiar enough to say. --Nanite (talk) 23:38, 3 March 2016 (UTC)
@Nanite I'm sorry. The starting point of statistical mechanics is the equal a priori probability postulate, i.e. that an isolated system (in absence of any further integrals of motion) is described by a microcanonical ensemble.
What you call an "illogical view", is nothing less than the foundation of statistical mechanics.
Before we discuss any of the other things we have to agree that isolated system are described by the microcanonical ensemble.
Do we agree on that? Davidwuelfert (talk) 09:32, 10 March 2016 (UTC)
To me, an interested bystander, it's fairly clear that the two of you do not in fact agree on that. --Trovatore (talk) 19:10, 10 March 2016 (UTC)
Indeed, I would rather say that a system is described by whichever probability distribution (ensemble) reflects our knowledge of that system. If we happen to perfectly know the energy of the system but not know anything more, then a microcanonical ensemble would be appropriate. Isolation would be necessary to such a state of perfect knowledge to continue (without being disturbed by a random interaction) but the converse is not true. Isolation is not sufficient to decide which ensemble is appropriate.
Well perhaps we are anyway running a bit off course, this is wikipedia after all and the article should be the primary object of our attentions. At the moment the article does rely on one possible take on the MCE to analyze negative temperature, and as David points out, many have raised good arguments why things are not necessarily so simple. I am just here to say that, on the third hand, we have the CE where negative temperature is somehow sensible and unambiguous. The article sort of switches to this when discussing lasing, but not so well. --Nanite (talk) 22:51, 13 March 2016 (UTC)
@Nanite No, we are not running off course. This article is called "Negative Temperature" and there is a number of published papers (see above) that (i) prove rigorously that negative temperatures are inconsistent with thermodynamics, the macroscopic theory that statistical mechanics tries to explain and (ii) show that the resolution for the apparent the conflict is using the volume entropy instead of the surface entropy. We got to discussing the foundation of statistical mechanics, because you argued that there was even a problem with the microcanonical ensemble itself. Moreover, you seem to disagree with the starting point of statistical mechanics, the equal a priori probability hypothesis and also seem to favour some theory that regards temperature as a more fundamental variable than energy. There is no such animal in physics as a negative temperature, that's what Dunkel and Hilbert showed in their Nature Physics paper. Davidwuelfert (talk) 12:29, 15 March 2016 (UTC)

## Applications?

So ... I guess the big question is: What applications do negative temperature materials have? Is there any specific interest in finding more of these materials, or is it just a peculiar property that some materials have? If they have applications in some areas, we should probably add a section to the article where we mention a few of them. —Kri (talk) 14:00, 24 July 2014 (UTC)

See the beautiful article population inversion. The electron shell system there is not at equilibrium though. 89.217.0.204 (talk) 09:01, 6 February 2015 (UTC)

## Population inversion

Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position.

Is this true? See population inversion article, which says this is impossible with just two states, at least in the case of electron energy levels about an atom. Or does that apply here? What is the mechanism of the "radio frequency" methods mentioned in the article? 89.217.0.204 (talk) 09:00, 6 February 2015 (UTC)

This is true. The case of electron energy levels does not apply here. The mechanism of the radio frequency method is the application of a radio pulse at the Larmor frequency of the specific nuclide for a sufficiently long time to cause an inversion. I will add an appropriate reference. 134.190.158.144 (talk) 14:17, 17 June 2015 (UTC)

## I do exist. I do, I do, I do!

Had to laugh. Several comments were made as to the non-existence or bee-ess-ness of this topic. I suppose it would appear a bit hard to understand to the non-physics majors amongst us (myself included). However, after reading a couple articles it is apparent that it does indeed exist but probably only in that lofty realm of quantum physics (where all bets are off for the time being). Perhaps it will be the basis of transporter devices of the future! LOL Anyhow, a few links for those pondering edits to this and related articles:

E. T. Janes' 1964 paper, Boltzmann vs. Gibbs Entropies, published in American Journal of Physics in 1965.[1]

What is the conceptual difference between Gibbs and Boltzmann entropies? [2]

Gibbs Entropy, Negative Energy - discussion on Physics Forum [3]

The Thomas Oikonomou -- the Physicist not the Greek actor -- paper on Clausius vs. Boltzmann-Gibbs Entropies. [4]

1. ^ "Gibbs vs. Boltzmann Entropies" (PDF). Retrieved 2 August 2016.
2. ^ "What is the conceptual difference between Gibbs and Boltzmann entropies?". Physics Stack Exchange Q&A. Retrieved 2 August 2016.
3. ^ "Boltzmann vs. Gibbs Entropy, Negative Energy". Physics Forums. Retrieved 2 August 2016.
4. ^ "Clausius vs. Boltzmann-Gibbs Entropies (2014)". Cornell University Library. Retrieved 2 August 2016.