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I saw this Nature article today, which cites e.g. arXiv:1211.0545.

And it makes no sense to me. The temperature of a collection of particles is the average kinetic energy of those particles. Kinetic energy cannot be less than zero (as far as I'm aware), so I don't understand what this article is trying to say, unless they're playing around with the conventional definition of "temperature".

The only thing I can thing of is if you have something like:

$$\frac{1}{kT} ~=~ \left(\frac{\partial S}{\partial E}\right)_{N,V}$$

And they've created a situation where entropy decreases with increasing energy.

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The relevant reference –  twistor59 Jan 3 '13 at 21:18
Possible duplicates: physics.stackexchange.com/q/21851/2451 and physics.stackexchange.com/q/38907/2451 and links therein. –  Qmechanic Jan 3 '13 at 21:28
This "The temperature of a collection of particles is the average kinetic energy of those particles." is very robust for a lot of systems but is not well defined for, say, Ising models. Using a maximally general defintion makes it possible for some cases, but as the link @Qmechanic provided notes it happens at high, not low energy in the system. Weird, but real. –  dmckee Jan 3 '13 at 21:39
I think if this is a duplicate of anything on this site, it would be this one. cc @Qmechanic. Nick, leave a comment if you find that one of these linked questions has the answer you're looking for. (And if not, could you edit this question here to explain how what you're asking differs from what has been asked before?) –  David Z Jan 3 '13 at 22:09
Honest question - why is this in the news all of a sudden? I was under the impression negative absolute temperature was a fairly pedestrian phenomenon. –  Richard Terrett Jan 5 '13 at 15:09

4 Answers 4

Your hypothesis that

if you have something like: $$\frac{1}{kT}=\left(\frac{\partial S}{\partial E} \right)_{N,V}$$ And they've created a situation where entropy decreases with increasing energy.

is exactly right. The concept of negative absolute temperature, while initially counterintuitive, is well known. You can find a few other examples on Wikipedia.

In your question you say that temperature is "the average kinetic energy of ... particles". Strictly speaking, this is only true for an ideal gas, although it's often a good approximation in other systems, as long as the temperature isn't too low. It's slightly more accurate to say that temperature is equal to the average energy per degree of freedom in the system, but that's an approximation too - energy per degree of freedom would be $E/S$, whereas $T$ is actually proportional to $\partial E/\partial S$, as you say. It's much better to think of $\partial E/\partial S$ as the definition of temperature, and the "energy per degree of freedom" thing as an approximation that's useful in high-temperature situations, where the number of degrees of freedom doesn't depend very much on the energy.

As Christoph pointed out in a comment, the significance of the new result is that they have achieved negative temperature using motional degrees of freedom. You can read the full details in this arXiv pre-print of the original paper, which was published in Science.

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the newsworthy part is that they achieved negative temperature via motional degrees of freedom –  Christoph Jan 4 '13 at 14:02
@Christoph yes, sorry, I didn't mean to diminish the significance of this work. I haven't read the full paper yet but it sounds like some pretty cool stuff. –  Nathaniel Jan 4 '13 at 14:28

A recent paper (ironically in Nature again) explains that negative temperature is a concept based on an inconsistent definition of entropy:

Dunkel, Hilbert (2014): Consistent thermostatistics forbids negative absolute temperatures:

The authors claim, that if one uses a consistent definition of entropy (the one from Gibbs) negative temperatures are not possible. So you are not the only one who thinks negative temperatures make no sense.

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I may be stupid but this paper really puzzles me. If anyone is interested in discussing the consequences this new definition would have on stat mech as a whole, I would gladly open a chat or something like that on the subject. –  gatsu Jan 14 '14 at 20:12

A simple answer is that a negative temperature can occur when one has an upside down Boltzmann distribution. Normally higher energy levels are never more populated than lower ones. But it is possible to force more systems into upper levels. One way is to align spins in a magnetic field and then reverse the field. Another is by laser excitation. When this happens, the formal temperature is negative. But note that this is a metastable state and as soon as the constraints maintaining it are removed, the system returns to "normal".

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That definition of temperature being average kinetic energy of all particles is colloquial. The information contained in specifying the temperature of any system is far more than what can be inferred from knowing the kinetic energy of all individual particles (if we can ever do so). Do realize that Temperature is a statistical concept while K.E. is not, there is no sense in saying the temperature of an individual entity in a system is something or temperature of a group of entities is something, the number of entities constituting in a system has to be large enough (of avogadro's number order) for the definition of temperature to hold. Going to second part of problem, Let $\Omega(S,V,N)$ be the number of microstates a system can take and $\epsilon$ be the energy.By classical statistical mechanics, definition of inverse temperature (which is $\frac{1}{\kappa T}$ is; $$\frac{\delta ln(\Omega)}{\delta \epsilon}$$ This is same definition as defined in thermodynamics. And more importantly, in statistical mechanics, temperature is a function of energy, $\epsilon$. Now, there is nothing that says, $\Omega$ has to be a monotonically increasing function of $\epsilon$. Imagine a situation where there are lots of energy levels and then an upper bound level. So the number of states would go up and then come down making the slope negative and hence the temperature.

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