[scottmsul]

The two-state system was the first example I learned in undergrad stat mech, and really helped me understand the first-principles definitions of entropy, temperature, etc. My parent comment was more for the casual HN reader, but if you really dig into the two state system, negative temperatures aren't all that weird.

The first thing to understand is microstates, which is just the number of ways a system can have a certain energy. Eg in a two-state system with ten particles and energies +/-(E/2), there's one microstate where the energy is -5E (all negative), ten states with -4E (one elevated), etc. Then entropy is just the log of the # of microstates, which is much easier to deal with, since microstates tend to behave exponentially. Eg entropy(E=-10) is log(1)=0, entropy(E=-9) is log(10), etc.

Then temperature like you said is d(entropy) / d(energy). Two systems with different temps brought into contact will exchange energy until the temps are equal, since this configuration maximizes entropy.

The two-state system can have negative temperatures since entropy starts decreasing with energy once more than half of them are in the higher-energy state. This can't happen in more familiar scenarios (eg ideal gas, blackbody, etc) since usually entropy always goes up with energy.