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Negative Temperature

The ``boundedness'' of  U  and the consequent ``peakedness'' of  $\sigma(U)$  have some interesting consequences: the slope of  $\sigma(U)$ [which, by Eq. (10), defines the inverse temperature] decreases steadily and smoothly over the entire range of  U  from  $-N \mu B$  to  $+N \mu B$,  going through zero at  U = 0  and becoming negative for positive energies. This causes the temperature itself to diverge toward  $+\infty$  as  $U \to 0$  from the left and toward  $-\infty$  as  $U \to 0$  from the right. Such discontinuous behaviour is disconcerting, but it is only the result of our insistence upon thinking of  $\tau$  as ``fundamental'' when in fact it is  $1/\tau$  that most sensibly defines how systems behave. Unfortunately, it is too late to get thermometers calibrated in inverse temperature and get used to thinking of objects with lower inverse temperature as being hotter. So we have to live with some pretty odd properties of ``temperature.''

Consider, for instance, the whole notion of negative temperature, which is actually exhibited by this system and can actually be prepared in the laboratory.^15.22 What is the behaviour of a system with a negative temperature? Our physical intuition, which in this case is trustworthy, declares that one system is hotter than another if, when the two are placed in thermal contact, heat energy spontaneously flows out of the first into the second. I will leave it as an exercise for the reader to decide which is most hot - infinite positive temperature or finite negative temperature.