[rotskoff]

Statistical physicist here. Negative temperatures occur when a system has a finite number of high energy states. On average, when temperature increases, both the energy and the "randomness" or entropy of a configuration increase as well. Of course, if there are only a few high energy states available, then the randomness will not increase, it will decrease. That's negative temperature! Because we define temperature as the rate of change of the energy with respect to the energy, in systems like the one I described, this rate of change becomes negative.

[cbolton]

I find your explanation much better than the one at https://simple.wikipedia.org/wiki/Negative_temperature, please consider expanding that page a bit!

Back to the substance of the topic: I feel let down here, negative temperature sounds like something amazing but it turns out to be more of a quirk due to the definition. I wonder if physicists would have chosen this definition, if had they been aware of this when they did.

Also I wonder if there's an intensive thermodynamic property that actually says how much thermal energy is in the system, since temperature apparently won't do it?

[Rayhem]

> Also I wonder if there's an intensive thermodynamic property that actually says how much thermal energy is in the system, since temperature apparently won't do it?

Thermodynamic beta[1] does exactly that. If we consider temperature as "tendency to give energy away", then the scale starts at zero, heads out through positive infinity, comes in through negative infinity, and then stops at negative zero. I.e. if T_a > T_b > 0, then system a gives energy to system b. Then, if T_c < T_d < 0, then system d gives energy to system c.

Thermodynamic beta (really just 1/T, the "coldness" of a system) fixes this: if B_a < B_b anywhere on the number line, then B_b is colder than B_a.

[1]: https://en.wikipedia.org/wiki/Thermodynamic_beta

[akira2501]

Do you think the big bang could have been an "entropy population inversion" event? All high entropy states were occupied, so whatever event started the big bang caused the universe to dip into negative temperature and entropy and allowed our universe to form?

Or.. am I just a crank?

[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.

[Davidzheng]

Rate of change of entropy i presume is what you meant

[drdrey]

What’s an example of such a system?

[Rayhem]

Lasers do this. The essential idea is that an incoming photon (of some specified energy E) knocks a bit of energy (specifically also E) out of an excited particle which leaves as another photon. To make this happen, you need more particles in an excited state than in the ground state[1] which is exactly the same condition necessary for negative temperature.

[1]: https://en.wikipedia.org/wiki/Population_inversion

[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.