Tuesday, January 08, 2013

Nonsensical hype on negative temperatures

Off-topic cinema: I've seen my first 3D movie in a cinema today, The Life of Pi about a sensitive Indian teenager whose family moves to Canada with the whole zoo but the boy (believer in all the world's major religions) is the only one who survives on a boat, together with their tiger named Richard Parker. Top Russian actor Gérard Depardieu of Saransk starred as a racist cook on the ship. :-) It turned out that the glasses were some kind of color anaglyphs. The question how it's possible that I can see all RGB channels in both eyes despite the filters replaced my analogous question for circular polarization glasses. ;-)
Fox News and tons of other sources proudly announced that physicists in Munich have finally realized temperatures colder than the absolute zero, –273.15 °C. Their article has ignited a storm in the Jan 4th issue of Science.

When you combine objects with people or worlds that are upside down, like in this 2012 Canadian-French romantic movie "Upside Down" about the love between Adam and Eve from two co-existing worlds, things may get confusing or, more often, downright inconsistent. One has to be careful. ;-)

Needless to say, I completely agree with David Berenstein:
Bad science reporting versus good science reporting
David picked Ars Technica as his example of the most acceptable reporting about this story.

What's going on?

First, we must ask: What does it mean to have a well-defined temperature \(T\) in kelvins (which we call the "absolute temperature")? Recall that the temperature \(t\) in Celsius degrees is linked to the absolute temperature \(T\) by an additive shift,\[

T = t+273.15 \,{\rm K}, \quad t=T-273.15\,{}^\circ {\rm C}

\] As a unit of the temperature difference, one kelvin and one Celsius degree are the same thing.

Roughly speaking, having a well-defined temperature means that the system carries the energy \(kT/2\) per degree of freedom (e.g. a coordinate of a spherically symmetric atom). Because one can't just "count" the relevant degrees of freedom in the most general situation (they're picked from an infinite bath of candidates but one should only pick those that have a chance to be excited, e.g. those with \(kT\geq \hbar \omega\), and things get hard when the formula for the energy is highly non-linear), we need a better, more universal definition.

The most universal definition of temperature is due to Boltzmann. The system has temperature \(T\) if the probability of the arrangement (microstate) \(n\) of the physical system is given by\[

p_n = C \cdot \exp\zav{ -\frac{E_n}{kT} }.

\] Here, and I should have already told you, \(k=1.38\times 10^{-23}\,{\rm J/K}\) is the Boltzmann constant. The coefficient \(C\) must be independent of \(n\) and it is chosen so that the total probability of all microstates is \[

\sum_n p_n = 1.

\] You see that if you have some temperature, the probability of states with a higher energy (imagine the kinetic energy of molecules of gas) exponentially decreases with the energy. The lower temperature \(T\) you have, the larger \(\beta\equiv 1/kT\) is, and the faster the decrease with the energy \(\sim \exp(-\beta E_n)\) is. In the extreme limit \(T=0\), the probabilities decrease "exponentially quickly" with the energy of the state which means that only the state with the lowest possible energy, the ground state, is allowed with the probability \(p_0=1\) (certainty).

Now, the condition \(\sum_n p_n=1\) may be easily satisfied – without encountering a divergence in the sum – because the number of available states isn't increasing exponentially with the energy so the exponential suppression cures any conceivable divergence for high energies. For low energies where the exponential is exponentially growing, we're saved by the fact that the energy is bounded from below. There exist no states with \(E\lt E_0\). For a fundamental enough theory (that can produce particles etc.) such as quantum field theory or string theory, \(E_0\) is the energy of the vacuum. There can't be any states whose energy is lower than the energy of the vacuum.

Things change if the temperature is negative. In that case, \(\exp(-\beta E)\) is exponentially increasing with \(E\) because \(-\beta\gt 0\) now. If there are states with arbitrarily high energies \(E_n\), it's clear that the sum of \(\exp(-\beta E_n)\) will diverge: the terms that you add are growing larger with \(E_n\) and arbitrarily large terms exist.

So the negative temperatures can't exist for a sufficiently general system. The condition that the "total probability of all alternatives equals one" can't be satisfied because the sum of the exponentially increasing probabilities inevitably diverges. It just can't be one.

However, one may find "negative temperature configurations", \(T\lt 0\), in restricted systems that effectively only have states for which the energy is bounded from below as well as from above. In that case (in which you have to manually prohibit changes to almost all the degrees of freedom and only allow e.g. one electron per atom to be excited or unexcited), the sum is convergent – in most cases, it's a sum of a finite number of terms, after all. There's still something interesting about the probabilities\[

p_n = C \cdot \exp\zav{ -\frac{E_n}{kT} }

\] if \(kT\lt 0\): these probabilities actually increase with \(E_n\). In other words, it's more likely that the physical system will be found in a state with a higher energy than it is that it will be found in a state with a lower energy. Well-behaved physical systems have positive temperature and they try to "save energy". They don't try to donate a gas molecule too much energy, for example. However, systems with negative energy exhibit the so-called population inversion. Much like the governments, they simply love to waste money (I mean energy). The more they can waste, the more likely it is that they will. ;-)

Now, it's been known for quite some time that lasers and masers are examples of population inversion. The materials that are expected to emit the nice laser light (or maser radiation) have to be brought to the state in which the excited atoms are more numerous than the unexcited ones. That's why they're ready to emit lots light quanta (and almost no one can absorb them) and why you ultimately get the coherent laser/maser light (you must understand the Bose-Einstein statistics for photons, too).

These negative-temperature materials can't be brought into thermal equilibrium with a general enough system because for a general enough system, the partition sum doesn't converge at negative temperatures because there are states with arbitrarily high energies, as I have mentioned. Instead, what happens to these negative-temperature systems when they're in touch with ordinary positive-temperature matter is described in textbooks on lasers, including the Wikipedia article above.

The lasers and masers have been known for quite some time – and awarded by a Nobel prize 49 years ago – so they're not new inventions. The realization that the population inversion indicates a negative absolute temperature has been known pretty much since the beginning, too. (For technical reasons, the negative temperature was first measured in masers, not lasers.)

So whatever the folks in Munich discovered, it is not a new and revolutionary breakthrough that brings us below the absolute zero for the first time. What they did was some incremental research of phenomena that may be described by the problematic yet catchy term "negative temperatures" but it's simply wrong if this "sexy X factor" is being overhyped in this revolutionary way. Their work simply isn't revolutionary. It is about "just another system" – the Bose Hubbard model – where the population inversion was achieved for new effective degrees of freedom, some motional ones (the momentum is effectively bounded/periodic because the model is defined on a lattice; that's why the negative-temperature construction isn't spoiled by an unlimited kinetic energy). But the thermodynamic essence isn't really different than in the case of lasers and masers.

(Needless to say, this overhyping of catchy terms is being done in many disciplines and I would say that high-energy physics is no exception. The laymen – including most of the sponsors – just can't follow the actual contemporary research so this research is being constantly justified by catchwords that have already existed for 50 years. But for some reasons, these concepts are still being claimed to be hot and new and exciting.)

While the possibility to obtain negative-temperature systems is, with all the necessary disclaimers, demonstrably real, the speculations that the negative-temperature systems are relevant as a description of dark energy is insanely speculative and, as far as I can say, it is wrong and won't be accepted by competent physicists.

To summarize, these stories combine some wrong speculative research about dark energy with "apparent oxymorons" that can be partly realized in practice but that have been realized for half a century so they shouldn't be described with these sensational "new and revolutionary labels". As far as I know, Ars Technica is the only source whose reporting of this story is tolerable – I am still not quite satisfied with it – and everyone else has failed miserably. If you realize that the reporting of this research has pretty much nothing to do with the truth and out of 3 fundamental mistakes about the interpretations, almost all the media pick all 3, you should feel at least as frustrated about the quality of the science-focused inkspillers as I do.

And that's the memo.


  1. nice articIe!.. I have a simple (and dumb) question:
    is it theoretically possible to have "hot negative temperature" configurations?
    i.e. high energies bounded from below and above and with some increasing probability density..
    that would be interesting -.-

  2. Thanks for your compliment and interest!

    I don't understand your question - in what sense what you proposed differs from what I have discussed.

    Incidentally, the adjective "hotter" is subtle for negative temperatures. For positive temperatures, T1 is hotter than T2 if T1 is greater than T2 or, equivalently, 1/T1 is smaller than 1/T2.

    For comparisons of positive and negative temperatures, the rule with the inversion leads to different answers. By comparing -5 K and +5 K, we may say that +5 K is warmer than -5 K. However, the inverse temperatures are +1/(5 K) and -1/(5 K). The latter is smaller, because it's negative, and smaller inverse temperatures were called "hotter" in the previous paragraph, so we should say that negative temperatures are, in some sense, hotter than any positive temperatures! ;-)

  3. In some sense negative temperatures are "higher" than any positive temperature, because when the system with the population inversion equilibrates with a normal system, it will give off heat in reverting to a normal positive temperature.

  4. yeah, I was just inventing some "paradoxical" definition :)
    This experiment concerns only ultracold atoms. It would be fun to get high energy negative systems, just to say "hot negative energy" :P

  5. Right, that's a nice operational explanation why.

  6. I like this nice reminder about well known systems with population inversion too :-)

    I have been frustrated about the science-focused inkspillers since a long time ago. If things go well, they just hype well known (as in this case) or wrong things, and in the worst case they try to start flame wars about certain (high energy) topics :-/.

    Thats why I immediately open my computer and klick TRF and other nice and friendly physics blogs if I want to learn about "what is shaking" in physics :-D


  7. Off topic: Did you like the movies? (The Life of Pi and Upside Down) I'm thinking to watch them but I don't know if they are good.

  8. Good question. I haven't seen Upside Down yet - it was my alternative plan but it will only be screened at the end of January and it's not 100% guaranteed that a 3D version will be ready at that time although they plan to see it. So I only know the trailer.

    Life of Pi was a good movie. I wanted something that would make the 3D effects impressive - a tiger running towards me etc. I think it was a kind of promotion of religions in general but if one ignored the deep religiosity of the boy, he wouldn't lose too much, I think. His father was a hardcore atheist whose life was saved by the Western medicine, by the way. ;-)

    The survival rate with the tiger on the same boat was of course a bit unlikely but I don't mind if movies are somewhat stretched - what I hate is "socialist realism" with totally mundane events everywhere.

  9. In "Upside Down" isn't the girl called Eden (and not Eve) ?... ;-)

    Didn't you get a headache with the 3D?... I always do. I find it a strain on the eyes.

  10. Sorry for the off topic Lumo but I am really worried about this:


  11. I agree with you. His question is troubling for two reasons. First, it is totally unnecessary since he should know that whether or not monopoles exist is an open question. I think they do (so does Polchinski), but primodial ones have been inflated out so are rare, and they are too massive to be produced by our accelerators.
    The other troubling aspect is suggesting some policy on open questions. Where do these controlling personalitlies come from?

  12. I agree. Monopoles haven't been observed, we don't know whether they exist or not, it's more likely that they do given the analysis of the relevant grand unified (and similar) theories.

    And it's an open question but not the only open question. It's silly to single out magnetic monopoles among open questions because open questions are inevitable in physics, pretty much by the definition of physics or science. I left a short comment of this sort over there.

  13. Thanks Lumo and Gordon,

    his attitude concerning open questions in theoretical and fundamental physics really scares me. Happily nobody seems to strongly agree with him up to now.

    Demanding to restrict such questions corresponds to bullying theoretical or fundamental physics. I fear that he will soon go after other topics, such as supersymmetry and string theory of course, too :-(((

  14. Darn, now Sklivvz is outright trolling about beyond the standard model physics :-(((, see his comments here:


    He demands that BSM physics, such as GUTs vor example, is accompanied by the adjective "speculative" as soon as it is mentioned, and his whole attitude towards BSM physics looks like if he is accusing scientists who are positive about it and say so in their answers are working against the scientific principle ect :-(((

    And all this he does even thoug you Lumo, Qmechanic, and even Dmckee who is an eperimental physicist, have explained to him why it is natural and legitimate for questions about open issues to have the spectrum of answers they have in different contexts.

    Boy, how he drives me up the wall again :-( !!!

  15. Dear Dilaton, I have added my own Answer there. As usual, though, I have my own tiresome and unwelcome commentary to give you also ;)

    You and the other physicists joined physics.SE when all the hard work to make it what is had already been done by others. Thousands and thousands of person-days have gone and are still going into the architecture, coding, design, and fine-tuning of Stackexchange. It is a commercial enterprise, not a community of, by, and for physicists[1]. According to Wikipedia, "[i]n March 2011, Stack Overflow raised $12 million in additional venture funding...". The investors who paid this $12 million did not do so out of philanthropic motives. They expect a handsome return on their investment and if they believe that enforcing a Disneyfied "I'm OK, you're OK" culture of "niceness" throughout their network is in their best commercial interest then nothing you say will sway them. In fact, they -- more precisely, the executives running the show -- have heard you loud and clear already, and the response as represented by shog9 is a hearty F U.

    Ron Maimon thinks that what Jeff Atwood, the principal force behind the development of Stackexchange, did was no big deal and that he mostly ripped off Usenet. Maimon also thinks that a rival to SE could be set up easily using free software. He is delusional. I am a user of ask.libreoffice.org, which runs on a free Stackexchange clone, and it is infuriating to use. It compares to Stackexchange like a teletype compares to a brand new computer.

    You happened to be there in a particular moment in the development of physics.SE when the culture was more freewheeling and more closely resembled the give-and-take, the cut-and.thrust found in actual discussions among scientists. I share your disappointment that this is ending but such is life. It's a big wide world and there are other outlets for your talents.

    [1] SE claims "Stack Overflow is run by you!" but that is for the gullible. It is run by the management and their staff, plus volunteer labor who must fall in line with SE management decisions or they will be removed.

  16. Thanks Eugene,

    I have already welcomed your nice answer and I appreciate your whole comment here ... ;-)

    Since the beginning of December last year I am some kind of cursing myself for not being able to let go and simply stop klicking that site :-(

    It partly is because it has been very useful and helpful and a great support in educating myself further about physics.
    The other issue is my darn naive hope that things will normalize and things will be ok again someday as time moves on.

    It is so darn hard to accept that all of the good things over there are definitely over and will probably never return :-(

    I always appreciate your thoughts (about many things).

  17. Dilaton--why are you playing nice with Eugene's nonsense? What sort of viable commercial enterprise can call itself physics.SE without the "give and take, cut and thrust found in actual discussions amongst scientists."? If that "free-wheeling" quality is removed, the "real" physicists (see the BBT) will also remove themselves, then you will have a physicist-free zone for shog9 and others to play in.

    Perhaps mathstack is more successful because non-mathematican mods keep their noses out of the site because the math is too technical for them to even begin to shut down or control questions.

  18. See, this (currently without a single downvote or comment other than my own) is why an enforced "niceness" culture that allows dilettantes to post stuff unchallenged is good for business but bad for the quality of information. I've never seen Ron use harsh words with a n00b/beginner but he could be quite abrasive when telling people claiming expertise (in my opinion, when you give an Answer you claim to provide an expert opinion) they were wrong and why. So he made enemies, PLUS the kind of people who get the vapors whenever they see somebody being "mean" got all upset... but together these two groups make up a large demographic!

  19. Exactly for those reasons I stopped all considerable effort on the site.
    Believe it or not: We are/were just content generators :)

  20. Dear Robert, isn't every work "content generation" of a sort? Farmers are providing with content for cans of luncheon meat, or whatever. ;-)

    From the website-managerial viewpoint, of course that users of websites are content generators. I don't see anything insulting about it. That doesn't mean that there aren't other viewpoints from which the work is important for other reasons.

  21. Yep this is blatant nonsense, now it has a downvote ... :-)

    I'd give my eyeteeth for having Ron (together with other good physicists) back and Chris Gerig or Ron's Hamster as a moderator (they run as mods too in the election), instead of having Manishearth together with his SE overlord friends and Sklivvz controlling the site now. They give a damn about the kind and level of content there is, as long as everybody is just nicely small talking. They are only interested in policying, inventing new rules, controll mechanisms, and restrictions they can enforce, annoying discussions about increasing the amount of policying etc.

    They should be concerned about the drop of incoming good questions we have and figure out how the level could be increased and the site be made attractive for physics experts again.

    I agree that TP.SE was and Math.SE was/is better because it was/is more technical. I guess when the nonphysics mod looked at the TP.SE site they had no clue what people there were talking about.

    That is why I keep regularly retagging some imported nice hardcore high level TP questions, it bumps them to the front page and should make them putting their noses out ;-P

  22. The shoking thing is that nobody (at least nobody who is vocal enough) seems to care about the kind and level of the content anymore.

    The level of the incoming new questions (since the escallation at the beginning of December) is on avarage freely rolling down to infinity, many still occuring good questions get only unsatisfactory answers from people who are not knowledgable enough in the topic the question is about... if they get answers at all, many fundamental physics topics such as the ones people on TP.SE had been interested, in are fading away since December last year too, etc ...

  23. That's finally it !

    Too many bad things are happening with an increasing frequency, such as decent answers: http://physics.stackexchange.com/a/15417 which started out upvoted as they deserve suddenly end with a negative score, people proudly copy mistakes read in popular articles and stubbornly refuse to accept it if they are told what is wrong about it: http://physics.stackexchange.com/q/49936

    So Sklivvz, Manishearth and all his nonphysics friends who have taken over the site and who are only interested in policying and give a damn about the physics can take it (to hell if they like...).

    I dont care any more, nothing can be done about it anyway. To bad, but not even Qmechanic, whom I like and appreciate so very much, can do anything about the increasingly negative tendencies in too many things.

    So I'll just keep retagging about 5 nice valuable TP questions per day (readingand them and the answers they have is fun) without looking at anything else on the site until I am through.

    Seing all the bad things happen tears me appart, saddens me to death and topics I am interested in and care about are fading away anyway ...

    So I wont miss anything ;-)

  24. All that is right. But a commercial enterprise that fails to supply what the consumer wants will fail. SE may need both quality and the rancor and it appears to be losing both. Of course it may need neither. I don't understand the business model. It costs nothing to participate. It allows one to create many accounts under any Id. It has little advertising.
    If it loses quality or becomes dictatorial I won't cost it any server bandwidth and I suspect it will fail. But I am not sure how it is to succeed.

  25. Their business model? Oh i don't know Robert. You could ask that question not only of SE but of any of the properties held by its investors. Twitter, tumblr, soundcloud, Disqus, DuckDuckGo... how do any of these expect to become a profitable enterprise? That may be not quite the right question to ask. How did Skype expect that they would ever earn a penny? The answer is, they never did. Yet Steve Ballmer's Microsoft paid $8.5 billion to its former owner Ebay. More than a year ago, Skype shows no sign of turning a profit, but that's not the seller's problem.

    SE won't fetch anywhere near as much as Skype in an IPO but I'd be very surprised if they don't collect a minimum of $100 million. Not bad for a $12 million investment, eh. If I may translate SE's mission statement for you from PR corporatespeak, it's to attract as many eyeballs with loyalty to their brand as possible by offering a network of slick Q&A websites.

    The problem, as you note, is balancing two conflicting demands. It's all very well to bill yourself as an expert knowledge forum: a place where physics researchers can ask each other about quantum entanglement, but real experts tend to lose patience after a while with software engineers and other dilettantes posing as experts on physics.SE. Yet if you allow the real experts -- the physicists -- to scare off the poseurs, there won't be enough people left to answer all the questions from high schoolers about balls rolling down inclines and other amateur inquiries. Fewer eyeballs = lower market value. It's a conundrum!

    Fortunately for the investors and founders (who I'm sure will likewise be handsomely rewarded in the IPO), they can borrow a famous line from John Maynard Keynes: "In the long run, we'll all be dead." In the short run, they'll become multi-millionaires if they aren't already.

    If I sound cynical and negative about SE, I'm not, really. In fact, I admire what the founders built and I respect the reputation of their investors. There are many reasons why someone is willing to give freely of their time to volunteer as an expert or wannabe expert on SE and they all deserve to be respected. Even if you look at it as nothing more than an MMORPG, with its silly gold badges and rep points -- and I know first-hand how addictive these are -- it still beats smoking a crack pipe and joining the Crips. Maybe SE should open an office in East LA...

  26. Thanks. Obviously I am no investor so I shall never have the kind of money that one can apparently make with such. I appreciate that this blog could not exist but for investments in Disqus.

  27. Sklivvz just cant stop trolling


    and nobody seems inclined to stop him either :-(

    Now David Zaslavsky has even deleted many of Lumo's very impportant comments on the other questions concerning the magnatic monopoles.

    BTW I am very disappointed about David, he did not the slightest thing to prevent the brutal disruption and blasting of the Physics SE community at the beginning of last December; he should have protected Ron and the whole community. And now, he is more interested in policying too, does not care about the physics of the site any more, and even supports trolling such as Sklivvz likes to do it.

  28. Zaslavsky does not have god-like powers, he is constrained by what SE management and staff tell him and by the general mood of the physics.SE users. The questions now are:

    (1) Is this recent change in physics.SE the result of a raid by three high-powered users from skeptics.SE (you know who these three are) who are working as a team to bully mild-mannered physicists and push them around or (2) actively encouraged by SE management based on the reasoning I have set out elsewhere on this page.

    In either case, Zaslavsky alone cannot do much. In case of (1), only a grassroots uprising by established physics.SE users can reverse the trend. If you want to see that happen, you should step aside now. You have fought the good fight and alerted others to the problem. If you continue agitating, you will be viewed as a troublemaker and your efforts will be counter-productive. If, however, (2) is the case, nothing at all -- short of a large-scale boycott -- will help.

  29. The supergau would be if both (1) AND (2) are true simultaneously, which is what I almost suspect ... :-/

    Yep, I have fought the good fight and clearly lost :-(
    It is really too badsad that I do not have the power to close/reopen vote, so all I was able to do is make noise.

    Otherwise, I could have closevoted all the bad attacks on legitimate posts by meta question Sklivvz attempts and voted to reopen wrongly closed good physics questions on the main page. This would have been much more efficient and less distractive for myself and anybody else...