[tom-thistime]

Take an atom that somehow only has two energy levels, 0 and 1. Connect the atom to a heat sink at temperature T. At very low T that atom almost always has energy 0. We know the energy of each atom very well, they're (almost) all 0. People say: there's very little entropy. At very high T the atom has expected energy nearly 0.5, and the probabilities of energy 0 or energy 1 are nearly equal. So that's maximum entropy. We're as ignorant as it's possible to be.

At negative temperature the expected energy of each atom is >0.5. But as the expected energy approaches 1.0, we know the energy of each atom very well. They're (almost) all 1. That's weird. It's weird enough that you can't assign a positive temperature to these atoms.

Physically, you can get to negative temperature by sneaking atoms into the 1 state. Pumping a laser is an example. But you can't get to negative temperature by just heating with finite-temperature heaters.

Entropy can be measured in bits. If we have 10 two-level atoms at extremely high temperature, that's 10 bits of entropy. The state might be 10 1100 1110 or 01 1101 0111 or any of 2^10 possibilities. On the other hand if we have 10 two-level atoms at extremely low positive temperature, the state is usually 00 0000 0000 and the entropy is close to 0 bits.

"Entropy" in bits is just the size of the random number you need in order to represent the system.