TITLE: How to achieve temperature below absolute zero?
QUESTION [1 upvotes]: Firstly i am not a physicist, so kindly go easy on me. So i grew up believing that absolute zero is the bottom of temperature chart. Recently i read that scientist achieved the feat of going below that mark. 
http://www.livescience.com/25959-atoms-colder-than-absolute-zero.html
According to what i know the molecules stop moving at absolute zero, and hence one cannot go past that temperature mark. 
So my questions are:

are these results are real or there is some other meaning they are implying?
How is it possible to achieve temperature below absolute zero?
theoretically what is the highest temperature one can reach, if the highest speed a molecule can reach is speed of light? (considering movement of molecules determine their temperature).

PS: I have read the related links/similar questions and duplicates, but couldn't understand it really. So please be kind enough to explain in layman terms, and please edit the question if there are any misconceptions.

REPLY [3 votes]: First of all: abandon the notion that temperature is proportional to (kinetic) energy. It is only valid in classical systems where the energy is mechanical. It is not a good way to define temperature.
So what is temperature? Two systems have the same temperature when they are in thermodynamic equilibrium. Otherwise heat will flow from the warmer to the cooler system until equilibrium is established. On a microscopic scale, transport of heat occurs by random processes. The condition for thermodynamic equilibrium is that the combined system is in its most probable state. It is the macroscopic state (described by macroscopic parameters like temperature and total energy) in which the number of microstates (all the different ways in which the positions and energies of atoms and electrons can be arranged that are compatible with the macrostate) is largest. Let us call this number Ω. It is a really gigantic number, normally very rapidly increasing with internal energy.
Now back to the two systems that are in thermodynamic equilibrium, let us call them A and B. It is reasonable to assume that Ω of the total system is the product of $\Omega_A$ and $\Omega_B$. When Ω of the total system is at its maximum for the distribution of energy between A and B, it won't change in first approximation when a small amount of energy is transferred from A to B. If this small transfer of energy increases $\Omega_A$ by 1 %, then $\Omega_B$ must go down by 1 %. 
So the general condition for two systems having the same temperature is that they have the same fractional change of Ω with internal energy. (We do not really need to mention entropy, but it is a logarithm of Ω and the requirement that both systems have the same derivative of entropy with respect to energy is the same condition.)
Now, in systems where there is a highest possible energy (for example spins in a magnetic field), Ω cannot always increase with energy. For high energies it must start go down somewhere. The fractional change of Ω with energy is then negative. This means that temperature is negative.
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