Quantum Velocity said:
i think heat must flow from the positive to negative for
Heat will flow from the hotter system to the colder one. That is the second law of thermodynamics. Actually, negative absolute temperature systems are hotter than positive ones, and are actually hotter than the infinite positive absolute temperature. You should take a look at the papers referenced in the link I gave below, it is pretty clear in there.
You have to think it this way: the definition for temperature can be given as:
##\displaystyle \left(\frac{\partial S}{\partial U}\right)_x=\frac{1}{T}##
But S is a concave function, this comes from thermodynamics (U is of course the internal energy). So to reach from positive to negative absolute temperatures, you must pass through a maximum of S:
##\displaystyle \left(\frac{\partial S}{\partial U}\right)_x=0\rightarrow T=\infty##.
With the system of spins is actually really easy to picture what happens. In the first place, you must have bounds in the energy so you can normalize the probability distributions.
If you place a system of spins in a magnetic field, the lowest energy state will be given with all the spins pointing in the same direction given by the external magnetic field. This is a zero entropy state, you have only one microstate for the given macrostate at T=0K. When you rise temperature, you will have some spins pointing in the direction of the external magnetic field, and others in the opposite direction, until you reach a maximum in entropy, a very disordered state, where there are many microstates compatible for the given macrostate at that temperature.
Now think of this situation, you start with the system in a positive absolute temperature state, close to T=0K, and suddenly you invert the external magnetic field: now the system will be trapped in a state of negative absolute temperature, with most spins pointing in the direction opposite to the external magnetic field. This is the population inversion that was mentioned before.
The thing is that the laws of thermodynamics and the usual statistical mechanics works in this negative absolute temperature states. You just change T by -T, and you see that the population of a given state will be:
##\displaystyle P_i \propto \exp{\left(\frac{E_i}{\kappa_B T}\right)}##,
so now the system tends to occupy the higher energy states, instead of the lower ones (which is the situation given by the sudden change in the direction of the external magnetic field).
