> Am I guessing correctly that this isn't at all related to negative energy?
Yes.
> When calling this a temperature below 0K - isn't this shoehorning the concept of temperature onto something it wasn't meant to do?
No. Suppose you're doing a Fluid Dynamics Simulation. You need to assign a temperature and pressure to each point in space to simulate gas flows. Using the negative temperature will correctly predict the evolution of the gas flow.
This is actually a strength of physics, having equations that can take on values that to your knowledge are unphysical (such as negative refraction indices for example) but still work. This means that when someone manages to figure out how to build such a system; we don't need to rework all of our theorems (conservation of energy; momentum; etc) to see which are still valid. We also have the ability to investigate the applications of such materials; should we ever be able to find a way to create them.
> Isn't this transition from a positive temperature to a negative temperature system non continuous, as in it needs to be set up with different properties
There are plenty of discontinuous transitions in thermodynamics, phase diagrams are an example. This is hardly an argument to that the concept of negative temperature is a broken one.
Thanks for your detailed response. Your example with fluid dynamics is an interesting one. If it really works as advertised and you can put these negative temperatures into every equation that takes a temperature, then yes, I can see the value. I guess I was just skeptical that it would actually work that way, since that's quite an amazing result ;-).
Regarding equations, for a physically unrelated but similar cognitive process, see[0] how we (A) set up a physical wave system, use complex numbers to work with wave equations, then liberally discard imaginary parts as if they never existed to get a physical result (B). (A) and (B) are anchored in physical reality, but the mathematical process in between is — as far as we know — not.
As for discontinuities, another non-thermodynamic physical discontinuity is the speed of light. Special relativity predicts it's entirely possible for particles to move FTL, when it's actually crossing the boundary that's a problem.
Yes.
> When calling this a temperature below 0K - isn't this shoehorning the concept of temperature onto something it wasn't meant to do?
No. Suppose you're doing a Fluid Dynamics Simulation. You need to assign a temperature and pressure to each point in space to simulate gas flows. Using the negative temperature will correctly predict the evolution of the gas flow.
This is actually a strength of physics, having equations that can take on values that to your knowledge are unphysical (such as negative refraction indices for example) but still work. This means that when someone manages to figure out how to build such a system; we don't need to rework all of our theorems (conservation of energy; momentum; etc) to see which are still valid. We also have the ability to investigate the applications of such materials; should we ever be able to find a way to create them.
http://en.wikipedia.org/wiki/Metamaterial
> Isn't this transition from a positive temperature to a negative temperature system non continuous, as in it needs to be set up with different properties
There are plenty of discontinuous transitions in thermodynamics, phase diagrams are an example. This is hardly an argument to that the concept of negative temperature is a broken one.
http://en.wikipedia.org/wiki/Phase_diagram