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.