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


*

*http://web.mit.edu/newsoffice/2013/its-a-negative-on-negative-absolute-temperatures-1220.html

*http://www.nature.com/nphys/journal/v10/n1/full/nphys2815.html

*http://arxiv.org/pdf/1304.2066v1.pdf
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.
A: 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".
A: 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. 
