Negative absolute temperature will burn your brain cells

The new year in science is so far shaping up to be a confusing one, with the first surprising physics result being the reaching of temperatures below absolute zero.

If that sounds bizarre to you, well it does to me too; but it’s thermodynamics and I’ve always found that difficult. But having calmed down and taken a bit of time to read and think about it, I think I can write something sensible.

What’s happened is that a team of German physicists have manipulated ultracold potassium atoms using lasers and magnetic fields to achieve a state that technically has negative temperature (Braun S, Ronzheimer JP, Schreiber M, Hodgman SS, Rom T, Bloch I & Schneider U 2013, “Negative absolute temperature for motional degrees of freedom”, Science, vol. 339 no. 6115 pp. 52-55, doi: 10.1126/science.1227831).

This works because temperature isn’t exactly what you think it is. We normally think of it as representing the average kinetic energy of the atoms or molecules in a sample, i.e. how rapidly they’re all jiggling around. So absolute zero corresponds to when they’ve stopped moving altogether, which is why it’s an absolute.

Well that’s almost right. The technical definition relates temperature to the distribution of energy in the system, and the rate that the entropy changes when the energy changes.

Entropy is an expression of the disorder in a system. You may remember it from the Second Law of Thermodynamics. That’s the one that says that the entropy in a closed system can never decrease: everything tends to move towards a more disordered state.

This is the reason that a cup that falls off a table won’t spontaneously put itself together again and jump back up. For one thing, it would need to convert heat energy – the random jiggling of atoms – into uniform motion of the whole cup. And of course the random jiggling is more disordered, i.e. has more entropy.

At least, that’s how it works at positive temperatures. Adding more energy increases the jiggling and increases the disorder. And until they receive more energy, most of the atoms can be found in the low energy, low entropy states.

Graph of entropy between minimum and maximum energies
Graph showing how entropy varies when there are both minimum and maximum energies. The temperature is actually the inverse of the slope of the curve, so it’s -0 at minimum energy and +0 at maximum energy. In the middle, where the entropy curve flattens out, it switches from positive to negative infinity (Image by Braun et al, Science)

But in this latest experiment they turned that around. First of all, they cooled the atoms down to a few billionths of a Kelvin. The atoms would normally repel each other, but they used lasers to trap them in a lattice arrangement.

Then they flipped it over. They changed the force between the atoms to an attractive one, and used the lasers to try to push them apart.

The result is that all the atoms were suddenly in a maximum energy state, but a very ordered one because they were all in this lattice arrangement. Any lower energy states were more disordered.

This reversed the relationship between energy and entropy, and so corresponds to a negative temperature.

Not only that, but because the Second Law of Thermodynamics means systems want to increase their entropy, it also means that the atoms want to lose their energy and move to a lower energy state.

So if you put a negative temperature system in contact with one with positive temperature, energy will tend to flow from the negative to the positive temperature. This has led some people to claim that negative temperatures are hotter than any positive temperature. Even though they’re a few billionths of a Kelvin below zero.

The way to understand this is that the temperature scale also doesn’t work the way you think it does. Normal number systems go from negative infinity, through zero and up to positive infinity. Like this:

−∞ … 0 … +∞

But with temperature, it works more like this:

+0 K … +∞ K, −∞ K … -0 K

So absolute zero is still a limit, it’s just a limit at either end. And the infinities meet in the middle.

I’ll let you go away and think about that, but leave you with one very cool (sorry) consequence.

Pressure and temperature are proportional to each other, so a negative temperature should also have negative pressure. And negative pressure is the very strange property exhibited by dark energy, which causes the accelerating expansion of the universe. It’s possible that by studying these weird, idiosyncratic atomic systems, we may get a better idea of how the cosmos works.

So it’s strange stuff, but worth understanding. If you still don’t get it and would like to read more, with clever analogies, see What the Dalai Lama can teach us about temperatures below absolute zero (Empirical Zeal), or Leprechauns and laser beams (Coffeeshop Physics).

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