What’s Cooler Than Being Cool?

You may have seen some news reports over the last week or two saying that scientists had made a substance with the hottest temperature ever recorded — but that temperature was somehow below absolute zero, a negative temperature on the Kelvin scale. Weirdly enough, this is absolutely true. A lot of the other stuff that showed up in those stories was completely false — my favorite was a statement that this would let us build a 100% efficient engine, breaking the laws of thermodynamics — but there is such a thing as a negative absolute temperature, and those negative temperatures are hotter than any positive temperature. In fact, these scientists pushed a substance a few billionths of a Kelvin below absolute zero, which is far and away the hottest temperature ever recorded. But the surprise is that they managed to do it with this substance in this particular way, not that negative temperatures are so hot. The idea of negative absolute temperature has been around for decades, and this isn’t the first substance to be prodded into a negative temperature.

So what is “negative absolute temperature”?

High school physics and chemistry teach us that temperature is a measure of molecular motion, or molecular kinetic energy: the faster the molecules in a substance are moving, the hotter that substance is. Contrariwise, as you make something cooler, its constituent molecules move more and more slowly, until they stop moving altogether, at which point you’ve reached absolute zero, 0 Kelvin.

This definition of temperature worked pretty well until quantum mechanics showed up in the early 20th century. (Quantum mechanics: screwing things up for a hundred years.) The problem is that, according to quantum mechanics, nothing ever really stops moving. This falls directly out of the Heisenberg Uncertainty Principle, which says that you can’t measure both a particle’s position and its momentum to arbitrary precision.1 If you got something to stop moving altogether, you’d know its momentum with perfect precision and its position with some finite precision, and that’s Not Allowed. So quantum mechanics says that, in general, molecules still have kinetic energy when they hit absolute zero — this is the “zero-point energy” that you may have heard about elsewhere (and that’s another subject that’s prone to silly & overblown reporting with breathless statements about breaking the laws of thermodynamics).2 This is a problem, though, because now we need a new definition of “absolute zero” — if it’s not the temperature where things stop moving, then what is it?

The best way to preserve the idea of “absolute zero” while still remaining consistent with quantum mechanics is a pretty intuitive re-definition: absolute zero is just the temperature where all the heat energy that you can possibly get out of the system has been taken from it; i.e. the molecules have as little kinetic energy as they can possibly have. But this minimum energy, the zero-point energy, varies from substance to substance — a molecule in a brick will have a different amount of zero-point energy than a molecule in a block of ice, and so on. So you can’t just point to the kinetic energy of the molecules in a substance and “read” the temperature off of that, because different substances are working off of a different baseline of minimum kinetic energy per molecule. This means temperature can’t be a measure of the kinetic energy of molecules anymore. So quantum mechanics forces us to re-define “temperature” too.

What’s the new definition of temperature? Quantum mechanics famously states that the amount of energy a molecule (or atom, or electron, or whatever) can have is quantized — it comes in tiny packets of a particular size. The new definition for temperature relies on this. This is where things get a little tricky, and we need an analogy.

Instead of talking about the kinetic energy of a collection of molecules, let’s talk about a bunch of rock climbers climbing up a cliff face. Better yet, there are many cliffs, each with lots of climbers on them, and some are climbing up and and some are climbing down. The cliffs are also infinitely high — climbers don’t actually reach the top, they just stop at some point and go back down. The climbers are our molecules, their height on their cliff is their kinetic energy, and each cliff corresponds to a diffImage credit: Mike Murphy. Licensed under the  Creative Commons Attribution-Share Alike 3.0 Unported license.erent object (a brick, a block of ice, etc.). You’re a cliff inspector (don’t ask), and your job is to figure out how hard each cliff is to climb just by looking at the positions of the climbers on the cliff face. So you decide to take the average height of the climbers on each cliff (the temperature). The harder it is to climb the cliff, the lower the average height should be (i.e. the colder the substance), so this should work well. And before that jerk Werner came along and screwed everything up, it did work.

When Werner showed up, he made your life (and the lives of the climbers) totally miserable. First, he made every cliff face totally sheer — no purchase anywhere at all. Then he attached a ladder to each cliff — but the ladders were all different. Some of them had huge spacings between their rungs, while some of them had tiny spacings; some had their first rung way off the ground, while others had their first rung nearly touching the ground (i.e. the zero-point energy varies from substance to substance). And while most of the ladders were infinitely long, just like the cliff faces, others weren’t — they just stopped partway up the cliff, so you couldn’t go any higher. But that wasn’t even the worst of it. Werner put some kind of spell on the climbers, forcing them to remain on the ladders for all eternity. They could never go lower than the lowest rung, and they could never leave the cliff faces.

So now you have a serious problem. You could still try to measure the average height of climbers on each cliff, but you’re not going to get the same information out of it. Some climbers are on ladders with the first rung way off the ground, and other climbers are on ladders that have their first rung very close to the ground, so it’s not fair to just measure their heights off the ground and be done with it. And since the distance between rungs also varies, you can’t just measure the average distance with respect to the first rung, especially if all the climbers are only on the first few rungs. ladderYou’d also like to find a way to be fair to the climbers stuck on finite ladders, who can’t just keep climbing up arbitrarily high, so average height definitely won’t work all around as a measure of difficulty. And, last but not least, you also want to be fair to the fortunate climbers on ladders with close spacings that have their bottom rungs really close to the ground. Since Werner hardly affected these special cliffs, you’d really like your new method to give basically the same answer for them that you got before Werner came along.

So what do you do? You decide that what really matters is how the climbers are arranged on the rungs. It stands to reason that the more climbers there are on lower rungs, the harder the ladder is to climb. Thinking a little more carefully, you decide that, in the simple case of a finite ladder that’s really easy to climb, every climber would reach the top, then turn around and climb back down, meaning that you’d have about the same number of climbers on every rung. So if a ladder that’s really hard to climb has everyone on the bottom rung, and a really easy ladder would have an equal number of people on every rung, all you have to do is figure out a way to “measure” between these two extremes, and you’ll have a new way of determining which ladders are easy to climb (hot) and which ones are hard to climb (cold). As it turns out that there’s a (relatively) easy way to do this with math, even though most of the ladders are infinitely long. So you try this out, and lo, it works! You can use the same formula for the finite and the infinite ladders, and surprisingly, when you use it on the special cliffs that Werner hardly affected, you get the same answer that you did before he came along. (In other words, the new quantum-mechanical version of temperature gives the same answer as the old-fashioned version of temperature when you’re not in the quantum regime.)

So the new definition for temperature (difficulty of climbing a cliff/ladder) relies on the arrangement of molecules (climbers) in their various possible discrete energy states (rungs on the ladder) that are allowed them by quantum mechanics (Werner, the jerk). If the molecules in a substance are all clustered down near their minimum energy (bottom of the ladder), then the substance is cold — but if they’re spread out more evenly among the allowed states (rungs), then the substance is hot.

Now we have a definition of temperature that plays nicely with quantum mechanics. Back to the original question: what is negative absolute temperature?

Back to the cliffs. Some of the ladders are finite. On those ladders, there really can be an equal number of climbers on each rung, the theoretical ideal of the easiest (hottest) ladder. If that happened, those ladders would be marked as “infinitely easy” (∞ Kelvin). But one day, when you look at one of the finite ladders, you notice something weird: the climbers have organized a conference at the top of the ladder. So there’s nobody at the bottom, and nearly everyone on the few rungs closest to the top. Therefore, your algorithm should say that while the conference is going on, the ladder is easier than the easiest possible ladder (i.e. hotter than ∞ K)!

Curious, you punch this arrangement of climbers into your algorithm, half-expecting your calculator to go up in smoke. Instead, you find something truly strange: the ease of this ladder comes out to be negative.

Going back to the real world: when there are more molecules near the top of the “ladder” of allowed energies than there are at the bottom, then a substance is hotter than ∞ K. For mathematical reasons, this means the temperature of the substance is said to be negative. What does this really mean? It means that if you artificially pump all the molecules in a substance up to their most energetic state, to the point where you have more molecules near the top than the bottom of the allowed range, then you’ve heated that substance  to a negative temperature.

Is there a reason this is called a negative temperature that goes beyond the mathematical? Sort of. The true definition of temperature, which comes from statistical mechanics, has to do with the relationship between energy and entropy (i.e. disorder). This sounds arcane, but in 99% of cases, this ends up reducing to exactly the same familiar definition of temperature as the average energy of molecular motion. As the molecules spread out across the ladder, occupying more rungs, the entropy of the system goes up, because there’s more disorder — stuff is spread out across the range of allowed energies, rather than clustered in one place. As you add more energy to the system, the entropy goes up.

Temperature is inversely proportional to the increase in entropy as you add energy to the system. When your system doesn’t have much energy, entropy is low because the molecules are all at roughly the same (low) energy. A little bit of extra energy goes a long way when there’s not much to begin with, so the entropy of the system goes up a lot as you add a little energy, and therefore the temperature is low (because it’s inversely proportional to the change in entropy). On the other hand, when there’s lots of energy in the system, the molecules have a wide range of energies, meaning entropy is high, and adding a little more energy only causes the entropy to increase a little bit. Therefore, the temperature is high. But if there’s a maximum energy — if the ladder has a top — and if all the molecules are clustered way up there near the maximum, then entropy is low because the energies of the molecules are clustered, and adding energy actually clusters the molecules’ energies more, decreasing the entropy. Since temperature is inversely proportional to the increase in entropy as you add energy, decreasing entropy means that you get a negative temperature.

Finally: how can there be a top to the “ladder” of allowed energies for molecules? The short answer is “quantum mechanics,” and the long answer is long. Very long. Well beyond the scope of this post. The important thing to remember is this: absolute zero is definitely the coldest possible temperature, but it’s not the lowest possible temperature — there are temperatures lower than absolute zero, all of which are intensely hot, due to the true nature of temperature.

  1. Put the philosophical issues with that off to the side: I have some qualms with that way of stating the HUP, but that doesn’t really matter for now. []
  2. Zero-point energy is also a real favorite among crackpots and charlatans, as you can see from the comments in the previous link. In fact, I’ll go out on a limb and predict right now that I’m  likely to get crackpot comments on this post simply because I’m using the phrase “zero-point energy” three times. []

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