[IgorPartola]

So the basic idea is that I’d you have a two particle system (a and b) and each particle can be in a low (a and b) or high energy (A and B) state then you can have the following combinations: ab, Ab, aB, and AB. Since particles are indistinguishable Ab and Ba are the high entropy states but ab and AB are low entropy states?

[Rayhem]

Basically. Instead of a and b, let's just look at L(ow) and H(igh) energy states. If I have the ground state,

    L L L L

there's only one way to arrange the system: everything in the low state, and the entropy is log(1) = 0. If I add one quantum of energy, I can have

    L L L H, L L H L, L H L L, H L L L

and the change in energy is dE = 1, while the change in entropy is dS = log(4) - log(1). "Temperature" is really just a scaling factor between dE and dS, so in this case T > 0. Adding another quantum,

    L L H H, L H L H, H L L H, L H H L, H L H L, H H L L

Again, dE = 1 (by construction), and dS = log(6) - log(4) which is less than log(4). dS > 0 so T is also greater than 0. Adding one more quantum, however,

    L H H H, H L H H, H H L H, H H H L

and we have dE = 1 and now dS = log(4) - log(6)! We've added energy (conventionally made it "hotter"), but the system has become more ordered. Adding one last quantum,

    H H H H

dE = 1 and dS = log(1) - log(4). This is as hot as the system can get, it cannot accept any more energy and can only give it away which is why "negative temperatures are hotter than all positive temperatures". If this system is brought into contact with any conventional positive temperature system, statistical fluctuations mean at least one of the energy quanta we added will flow towards it, cooling this system and heating the other.

[bp0017]

Thank you, this made a lot more sense!

[tom-thistime]

I wish I had more than one upvote.