Just to get the thoughts rolling…
Consider a two state system with discrete energy levels $E_1$ and $E_2$ where $E_2 > E_1$ which contains $N$ particles.
We can easily deduce that the state of maximum entropy is when energy level $E_1$ and $E_2$ each contain $\frac{N}2$ particles.
Based on the (flipped-upside-down) thermodynamic definition of temperature, $$(\frac{\partial E}{\partial S})=T$$
we see that when energy as a function of entropy increases, temperature also increases.
This means, for our two state system, every time a particle populates level $E_2$, there is a corresponding increase in temperature because that population increases entropy.
What about when we go beyond the point of maximum entropy where there are $\frac N2$ particles in $E_1$ and $E_2$?
Well, then we are adding energy to the system, but the entropy is decreasing.
Based on the Boltzmann factor, this can't be possible unless the temperature of the system is negative. That is, $$\frac{P_1}{P_2}=e^{-\frac{(E_1-E_2)}{kT}}$$
which approaches infinity unless one says the system has a negative absolute temperature.
Now we're getting to the point: this increase in energy which is connected to a decrease in entropy is what has been called a negative kelvin temperature. This is most often connected to population inversion in lasers.
Forgive me for saying things you already know.
All of this raises a lot of questions though, the most striking to me is that there does not seem to me to be a good physical explanation for how this can be possible. For instance, when people say that the negative temperature system is "hotter" I get what that means as far as saying that heat will flow from a negative temp. system to a pos. temp system.
I'm fine with that except that it invokes the idea of a fairly macroscopic quantity–hotness–for a purely quantum phenomenon. So, is there any macroscopic physical meaning to this statement that the negative temperature is "hotter"?
Concerning population inversion in lasers: doesn't this mean that the negative temperature required for that laser to exist ought to be so "hot" no mere mortal can handle such a device.
After all, negative absolute temperatures are often explained as being "hotter than infinite temperature." (I've heard this other places, but also see the neg. temp. Wikipedia article for quote.) I understand that again from a mathematical perspective, but what is the physical connection? A statement like that must surely manifest itself in some way that is noticeable beyond the quantum level.
I guess I really want to understand how it could be this "hotter than infinite temperature" system isn't perceptible in the macroscopic realm? I mean, lasers are cool, but they aren't just melting through everything as if they have infinite temperature.
My bad for the obvious misuse of the word "physical" when I probably meant something more like "macroscopic". I hope the point is clear anyways.
Thoughts?
Best Answer
If you go far up enough in the atmosphere, you'll reach regions where the temperature is over 3000 K. Yet rockets aren't exploding in flames when they get there, because the atmosphere is very sparse. It will transfer energy to the rocket, because its temperature is very high, but only incredibly slowly.
In general, human beings cannot detect temperature; we can only detect heat transfer. This is why metal surfaces often feel cold; they're the same temperature as everything else, but they conduct better, so you lose heat faster when touching them. Similarly, if you go up really high into the atmosphere, you won't feel like it's hot. Instead, you'll feel nothing at all.
So our intuitive sensory notions of "hot" or "cold" have very little to do with temperature. So it's indeterminate how "hot" a negative temperature system will feel, because that depends on how quickly it transfers heat to your body. All the temperature means is that it will transfer heat.