## Reading the Comics, August 3, 2018: Negative Temperatures Edition

So I’m going to have a third Reading the Comics essay for last week’s strips. This happens sometimes. Two of the four strips for this essay mention percentages. But one of the others is so important to me that it gets naming rights for the essay. You’ll understand when I’m done. I hope.

Angie Bailey’s Texts From Mittens for the 2nd talks about percentages. That’s a corner of arithmetic that many people find frightening and unwelcoming. I’m tickled that Mittens doesn’t understand how easy it is to work out a percentage of 100. It’s a good, reasonable bit of characterization for a cat.

John Graziano’s Ripley’s Believe It Or Not for the 2nd is about a subject close to my heart. At least a third of it is. The mention of negative Kelvin temperatures set off a … heated … debate on the comments thread at GoComics.com. Quite a few people remember learning in school that the Kelvin temperature scale. It starts with the coldest possible temperature, which is zero. And that’s that. They have taken this to denounce Graziano as writing obvious nonsense. Well.

Something you should know about anything you learned in school: the reality is more complicated than that. This is true for thermodynamics. This is true for mathematics. This is true for anything interesting enough for humans to study. This also applies to stuff you learned as an undergraduate. Also to grad school.

So what are negative temperatures? At least on an absolute temperature scale, where the answer isn’t an obvious and boring “cold”? One clue is in the word “absolute” there. It means a way of measuring temperature that’s in some way independent of how we do the measurement. In ordinary life we measure temperatures with physical phenomena. Fluids that expand or contract as their temperature changes. Metals that expand or contract as their temperatures change. For special cases like blast furnaces, sample slugs of clays that harden or don’t at temperature. Observing the radiation of light off a thing. And these are all fine, useful in their domains. They’re also bound in particular physical experiments, though. Is there a definition of temperature that … you know … we can do mathematically?

Of course, or I wouldn’t be writing this. There are two mathematical-physics components to give us temperature. One is the internal energy of your system. This is the energy of whatever your thing is, less the gravitational or potential energy that reflects where it happens to be sitting. Also minus the kinetic energy that comes of the whole system moving in whatever way you like. That is, the energy you’d see if that thing were in an otherwise empty universe. The second part is — OK, this will confuse people. It’s the entropy. Which is not a word for “stuff gets broken”. Not in this context. The entropy of a system describes how many distinct ways there are for a system to arrange its energy. Low-entropy systems have only a few ways to put things. High-entropy systems have a lot of ways to put things. This does harmonize with the pop-culture idea of entropy. There are many ways for a room to be messy. There are few ways for it to be clean. And it’s so easy to make a room messier and hard to make it tidier. We say entropy tends to increase.

So. A mathematical physicist bases “temperature” on the internal energy and the entropy. Imagine giving a system a tiny bit more energy. How many more ways would the system be able to arrange itself with that extra energy? That gives us the temperature. (To be precise, it gives us the reciprocal of the temperature. We could set this up as how a small change in entropy affects the internal energy, and get temperature right away. But I have an easier time thinking of going from change-in-energy to change-in-entropy than the other way around. And this is my blog so I get to choose how I set things up.)

This definition sounds bizarre. But it works brilliantly. It’s all nice clean mathematics. It matches perfectly nice easy-to-work-out cases, too. Like, you may kind of remember from high school physics how the temperature of a gas is something something average kinetic energy something. Work out the entropy and the internal energy of an ideal gas. Guess what this change-in-entropy/change-in-internal-energy thing gives you? Exactly something something average kinetic energy something. It’s brilliant.

In ordinary stuff, adding a little more internal energy to a system opens up new ways to arrange that energy. It always increases the entropy. So the absolute temperature, from this definition, is always positive. Good stuff. Matches our intuition well.

So in 1956 Dr Norman Ramsey and Dr Martin Klein published some interesting papers in the Physical Review. (Here’s a link to Ramsey’s paper and here’s Klein’s, if you can get someone else to pay for your access.) Their insightful question: what happens if a physical system has a maximum internal energy? If there’s some way of arranging the things in your system so that no more energy can come in? What if you’re close to but not at that maximum?

It depends on details, yes. But consider this setup: there’s one, or only a handful, of ways to arrange the maximum possible internal energy. There’s some more ways to arrange nearly-the-maximum-possible internal energy. There’s even more ways to arrange not-quite-nearly-the-maximum-possible internal energy.

Look at what that implies, though. If you’re near the maximum-possible internal energy, then adding a tiny bit of energy reduces the entropy. There’s fewer ways to arrange that greater bit of energy. Greater internal energy, reduced entropy. This implies the temperature is negative.

So we have to allow the idea of negative temperatures. Or we have to throw out this statistical-mechanics-based definition of temperature. And the definition works so well otherwise. Nobody’s got an idea nearly as good for it. So mathematical physicists shrugged, and noted this as a possibility, but mostly ignored it for decades. If it got mentioned, it was because the instructor was showing off a neat weird thing. This is how I encountered it, as a young physics major full of confidence and not at all good on wedge products. But it was sitting right there, in my textbook, Kittel and Kroemer’s Thermal Physics. Appendix E, four brisk pages before the index. Still, it was an enchanting piece.

And a useful one, possibly the most useful four-page aside I encountered as an undergraduate. My thesis research simulated a fluid-equilibrium problem run at different temperatures. There was a natural way that this fluid would have a maximum possible internal energy. So, a good part — the most fascinating part — of my research was in the world of negative temperatures. It’s a strange one, one where entropy seems to work in reverse. Things build, spontaneously. More heat, more energy, makes them build faster. In simulation, a shell of viscosity-free gas turned into what looked for all the world like a solid shell.

All right, but you can simulate anything on a computer, or in equations, as I did. Would this ever happen in reality? … And yes, in some ways. Internal energy and entropy are ideas that have natural, irresistible fits in information theory. This is the study of … information. I mean, how you send a signal and how you receive a signal. It turns out a lot of laser physics has, in information theory terms, behavior that’s negative-temperature. And, all right, but that’s not what anybody thinks of as temperature.

Well, these ideas happen still. They usually need some kind of special constraint on the things. Atoms held in a magnetic field so that their motions are constrained. Vortices locked into place on a two-dimensional surface (a prerequisite to my little fluids problems). Atoms bound into a lattice that keeps them from being able to fly free. All weird stuff, yes. But all exactly as the statistical-mechanics temperature idea calls on.

And notice. These negative temperatures happen only when the energy is extremely high. This is the grounds for saying that they’re hotter than positive temperatures. And good reason, too. Getting into what heat is, as opposed to temperature, is an even longer discussion. But it seems fair to say something with a huge internal energy has more heat than something with slight internal energy. So Graziano’s Ripley’s claim is right.

(GoComics.com commenters, struggling valiantly, have tried to talk about quantum mechanics stuff and made a hash of it. As a general rule, skip any pop-physics explanation of something being quantum mechanics.)

If you’re interested in more about this, I recommend Stephen J Blundell and Katherine M Blundell’s Concepts in Thermal Physics. Even if you’re not comfortable enough in calculus to follow the derivations, the textbook prose is insightful.

John Hambrock’s The Brilliant Mind of Edison Lee for the 3rd is a probability joke. And it’s built on how impossible putting together a particular huge complicated structure can be. I admit I’m not sure how I’d go about calculating the chance of a heap of Legos producing a giraffe shape. Imagine working out the number of ways Legos might fall together. Imagine working out how many of those could be called giraffe shapes. It seems too great a workload. And figuring it by experiment, shuffling Legos until a giraffe pops out, doesn’t seem much better.

This approaches an argument sometimes raised about the origins of life. Grant there’s no chance that a pile of Legos could be dropped together to make a giraffe shape. How can the much bigger pile of chemical elements have been stirred together to make an actual giraffe? Or, the same problem in another guise. If a monkey could go at a typewriter forever without typing any of Shakespeare’s plays, how did a chain of monkeys get to writing all of them?

And there’s a couple of explanations. At least partial explanations. There is much we don’t understand about the origins of life. But one is that the universe is huge. There’s lots of stars. It looks like most stars have planets. There’s lots of chances for chemicals to mix together and form a biochemistry. Even an impossibly unlikely thing will happen, given enough chances.

And another part is selection. A pile of Legos thrown into a pile can do pretty much anything. Any piece will fit into any other piece in a variety of ways. A pile of chemicals are more constrained in what they can do. Hydrogen, oxygen, and a bit of activation energy can make hydrogen-plus-hydroxide ions, water, or hydrogen peroxide, and that’s it. There can be a lot of ways to arrange things. Proteins are chains of amino acids. These chains can be about as long as you like. (It seems.) (I suppose there must be some limit.) And they curl over and fold up in some of the most complicated mathematical problems anyone can even imagine doing. How hard is it to find a set of chemicals that are a biochemistry? … That’s hard to say. There are about twenty amino acids used for proteins in our life. It seems like there could be a plausible life with eighteen amino acids, or 24, including a couple we don’t use here. It seems plausible, though, that my father could have had two brothers growing up; if there were, would I exist?

Jason Chatfield’s Ginger Meggs for the 3rd is a story-problem joke. Familiar old form to one. The question seems to be a bit mangled in the asking, though. Thirty percent of Jonson’s twelve apples is a nasty fractional number of apples. Surely the question should have given Jonson ten and Fitzclown twelve apples. Then thirty percent of Jonson’s apples would be a nice whole number.

I talk about mathematics themes in comic strips often, and those essays are gathered at this link. You might enjoy more of them. If Texts From Mittens gets on-topic for me again I’ll have an essay about it at this link.. (It’s a new tag, and a new comic, at least at GoComics.com.) Other discussions of Ripley’s Believe It Or Not strips are at this link and probably aren’t all mentions of Rubik’s Cubes. The Brilliant Mind of Edison Lee appears in essays at this link. And other appearances of Ginger Meggs are at this link. And so yeah, that one Star Trek: The Next Generation episode where they say the surface temperature is like negative 300 degrees Celsius, and therefore below absolute zero? I’m willing to write that off as it’s an incredibly high-energy atmosphere that’s fallen into negative (absolute) temperatures. Makes the place more exotic and weird. They need more of that.