Since T = dQ/dS, and S = k ln W, if you can create a system where adding energy reduces W, the number of ways the system can be in a similar configuration, then a positive dQ makes a negative dS.

Suppose you have a system of 50 magnets in a magnetic field. 49 of them are aligned against the external field, and 1 with. This represents an energized system, with 49Q and 50 possible states. To spin the 50th magnet needs 1 Q, and then there will be only 1 possible configuration. 49Q and 50 ways becomes 50Q and 1 way.

dQ = Q, and dS = k(ln(1) – ln(50)) = – k ln(50); and T = -Q/(k ln(50)).

If I’m not mistaken, it would be most correct to say “not moving relative to whatever is making the measurement of its position”.  (This is of course rather problematic, as being able to precisely locate the relative position of an atom would be an uncertainty principle violation.)

That blows my mind.

I don’t think zero molecular movement is possible (measuring the temperature requires probing it with a sensor; probing will require an exchange of energy, and this energy exchange will jiggle the molecules), but apparently it is possible in principle to have a collection of molecules whose beta value is zero.  Since temperature is the reciprocal of beta, though, such a substance would have an undefined (not zero) temperature. 

Based on Phanatic’s comment and my own limited understanding of this issue, substances with zero or negative beta values will actually be “hotter” (more energetic) than substances with similar positive temperatures, but don’t take my word on it.

Thanks for the explanation!

Just like with the supposed wierdness of quantum mechanics, the key insight is that the words don’t mean what you think they do. To a physicist, temperature isn’t “The thing a thermometer measures,” it’s “The derivative of energy with respect to entropy.”

Energy (traditionally written as U) = just what you think, measured in joules
Entropy (traditionally written as S) = the logarithm of the number of ways it is possible to divide the available energy up among the degrees of freedom present in a system.
Temperature = dU/dS

For most systems, this does what you’d expect. If you add energy to a gas, you can each each additional unit of energy to any molecule you want (making it go faster, vibrate more, ionize, whatever). So entropy always goes up when you add energy – and so dU/dS is always positive.

But put iron atoms in a magnetic field, and there are only 2 magnetic states – parallel to the field or opposed to it. The lowest energy state is every atom aligned, the highest energy states is every atom opposed. Both of these are states of zero entropy – there is only 1 arrangement of energy that achieves them. So to get from one to the other entropy must go up, then down as you add energy – meaning temperature (dU/dS) is first positive then negative. In practice you make such a system by aligning the magnet to an external field, then rapidly reversing the external field. Not a practical heat in engine, since that takes more work than the “bonus” you’d get out by using the result in a carnot-equivalent cycle. And actually, it isn’t clear that there *is* a way to make a carnot-equivalent engine for such a system.

So yeah, the Kelvin scale is defined in a way that doesn’t go from negative infinity to positive infinity. It looks qualitatively like a tangent curve: zero to positive infinity, jump to negative infinity, to zero.

Which, of course, really made me laugh when I played the Sims 3 and earned the “-1 Kelvin refrigerator.”