# Temperature below absolute zero?

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.

-
The relevant reference –  twistor59 Jan 3 at 21:18
Possible duplicates: physics.stackexchange.com/q/21851/2451 and physics.stackexchange.com/q/38907/2451 and links therein. –  Qmechanic Jan 3 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 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 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 at 15:09
show 1 more comment

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.

-
the newsworthy part is that they achieved negative temperature via motional degrees of freedom –  Christoph Jan 4 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 at 14:28