Good explanation. The main thing one needs to understand "negative
temperature" is the definition of temperature! One is sunk if one only
knows that temperature has something to do with how hot things are. :-)
Note that it's not *logically* necessary for a system to have a
maximum allowed energy, or finite number of states, to admit
negative temperatures. All one needs is that for certain ranges of
energy the number of available states decreases with increasing energy.
One can imagine systems for which S is a very wiggly non-monotonic
function of E, thus having lots of energies at which the temperature
becomes negative. Question: is there any simple way to design a
substance, or system, for which S is approximately a sinusoidal function of
E? I bet such stuff would have bizarre properties.
It's also worth noting from the 1/T in the formula above, that the
temperature rises to +infinity and then pops down to -infinity before
becoming negative. (Please, let's not get into the "is infinity for
real?" business again here...) This is why it's often said that
a system with temperature below absolute zero may also be regarded as
being hotter than infinitely hot! :-)
If I may now go off on a mathematical tangent plane...
one should really regard temperature as living on the projective line, that
is, toss in the point at infinity. One then has, instead of
the real line for a model of temperature, a circle --- just grab the two
ends of the line, +/- infinity, and glue them together.
In fact, it doesn't hurt to work in the *complex*
projective line, CP^1 (also known as the Riemann sphere).
Indeed, a very fruitful analogy between thermodynamics and quantum mechanics
arises from considering time, t, as i times reciprocal temperature,
1/T (which the cognoscenti prefer to call beta). Just as time can be positive
or negative, why not temperature? Anyone who's read Hawking's "A Brief
History of Time" will recall his description of "Euclidean time" as
being like a sphere, and indeed, this comes from his taking "complex time"
very seriously, which is a particularly daring maneuver in the context of
general relativity, but one also championed by Penrose.
Lest anyone think that this unification of time and temperature in the
context of the complex plane (or the Riemann sphere) is of interest only
to wild-eyed visionaries, I should also point out that the Kubo-Martin-
Schwinger, or KMS, condition for a state to be in thermal equilibrium is
based on this idea. Even humble workers in the subject of operator algebras
use KMS states these days.