The Clockwork Rocket

Yalda was surprised. “That sounds a bit extreme.”

Cornelio said, “If I told you that your theory implies that everything in this room is hotter than infinitely hot, would that justify rewriting the textbooks?”

“Infinity is my least favorite temperature,” Yalda confessed. “If you’re serious, I might have to recant.”

Cornelio buzzed softly. “Let’s call them negative temperatures, then; that’s formally correct, though the first way of speaking has its merits too.”

Yalda found the second way much more agreeable. “True energy has the opposite sense to kinetic energy, so to be consistent I suppose you could just declare all temperatures to be negative. Since a hot gas has less true energy than a cold gas, its temperature should be less… no?”

Cornelio was regarding her with an exasperated expression, but he was too polite to articulate precisely what he was feeling.

“I’m a physicist, show some mercy!” Yalda pleaded. “Thermodynamics is your domain. All I ever studied was the ideal gas law.”

“Temperature is not a synonym for energy,” Cornelio said sternly. “It’s about the proclivity of energy to pass from one system to another, not the quantity of energy that either one contains.”

“I’m willing to believe that,” Yalda said. “But how do you make such a ‘proclivity’ precise?”

“First,” Cornelio said, “think about the range of different ways in which one system can possess the same energy. Start with a single particle of gas, under the old physics.”

He summoned a diagram onto his chest. “The particle’s kinetic energy is proportional to its momentum squared. Pick a few examples of the energy the particle might have, but don’t pin it down precisely; just say that the energy lies in some small interval. From the plot on your left, you can read off a corresponding range for the momentum in each case.”

Yalda examined the diagram. “So you follow the horizontal lines for energy across until they hit the curve, then drop them down to the momentum axis?”

“That’s correct,” Cornelio said. “But then, recall that momentum is a vector. The energy has given us a range of sizes for that vector, but no information at all about its direction. The particle might be traveling north, west, up, down; we don’t know. So, take an arrow whose length you know, more or less, and swing it around freely, without any constraints. The tip of the arrow traces out a sphere—or rather, because the length isn’t fixed exactly, a spherical shell. The volume of that shell in ‘momentum space’ represents all the possibilities open to the particle, while still having an energy that lies within the given range.”

Yalda said, “So you’ve sketched parts of these shells, and plotted their volume against the kinetic energy… which turns out to be the same kind of curve as the momentum itself.”

“In this case, yes,” Cornelio said, “but that’s not true in general! So forget the resemblance, and just concentrate on the right-hand curve on its own terms. What does it tell you?”

“The volume in momentum space gets larger as you increase the kinetic energy,” Yalda said. “That makes sense. A faster particle has its momentum lying on a bigger sphere; the shells do get thinner as the momentum grows, but the larger surface area of the sphere more than compensates for that.”

“So the volume grows,” Cornelio agreed, “but when does it grow most rapidly?”

“At the start,” Yalda said. “When the energy is low, the volume shoots up; after that, it grows ever more slowly.”

“Precisely.”

“But where does that get us?”

“Particles bounce around, collide, exchange energy,” Cornelio said. “Give a particle a little more energy when its original energy is low, and the volume in momentum space accessible to it shoots up. And if it happens to get that energy by colliding with a particle that was moving faster, the volume for the faster particle goes down—but not by as much.”

“So… you need to add the two volumes?” Yalda suggested. “And see how the sum changes when energy moves from one particle to the other?”

“Not quite,” Cornelio said. “You multiply them. Each volume measures the possibilities that are available to one particle—and each possibility for one can be accompanied by any of the possibilities for the other. So it’s the product that you need.” He produced a new diagram.

“If energy moves from one system to the other, the product of their momentum space volumes grows along one edge of this rectangle, and shrinks along the other edge. So whether there’s an overall growth in the product depends on which of those changes is the larger.”

Yalda said, “You describe one system as being hotter and one colder—but where does temperature appear in all this?”

Cornelio said, “For each system, take the volume in momentum space and divide it by its rate of change with respect to energy. That encodes all the relevant information in a single number: the temperature. Then if one system’s temperature is greater than another’s—so long as they’re either both positive or both negative—that immediately tells you that if the first system gives energy to the second, it will increase the total number of possibilities. That’s why energy flows from hot to cold: the result ends up encompassing more possibilities.”

“Whew.” Yalda summoned her own version of Cornelio’s first diagram onto her chest and performed the final stage of the calculation. “So in our simplest possible example, temperature ends up being… proportional to kinetic energy! All that work, to get back to the naive idea that they’re really the same thing.”

Cornelio resisted rebuking her further. “Of course the true definition doesn’t contradict any of the results you were taught—for an ideal gas, under the old physics. But if you’re still clinging to some notion that temperature and energy are the same, take a look at what your own work has given us.”

Yalda gazed at his finely ridged skin, feeling suitably chastened and bamboozled. Then she began following the steps he had described for the simpler case, and the whole strange construction took on an eerie inevitability.

The true energy and momentum were linked by a circle, each simply rotating into the other. As the particle’s momentum grew from zero, its true energy began to fall—and at first, everything behaved very much like the earlier calculations, merely plotted upside-down.

But as the particle moved faster, its momentum could no longer increase without bounds. With the momentum levelling out, not only did the shells in momentum space cease growing so quickly, they became much thinner. At about two-thirds of the maximum total energy, the shells reached a peak in volume and began to shrink.

At that point, the effect of a change in energy on the number of possibilities open to the particle was reversed. A slow-moving particle could gain options by moving a bit faster… but a sufficiently fast-moving particle would lose options if it sped up. The ceiling on momentum made things cramped at the top.

The same thing showed up in the temperature, which switched sign when the shells’ volume peaked. And while negative temperatures on their own might merely have been the result of an idiosyncratic choice of conventions, Cornelio’s diagram made it clear that both negative and positive were real possibilities. You could always swap the labels for them by tinkering with the definitions, but you couldn’t banish the distinction itself.

Yalda said, “If everything in this room has a negative temperature, where are the positive ones?”

“On the surface of the sun,” Cornelio replied. “In our own burning stones.”

“I see.” A burning stone heated its surroundings, adding kinetic energy to them, so true energy would have to be flowing the other way, into the flame. Did that make sense? Cornelio had warned her that energy only flowed from the higher temperature to the lower if both had the same sign.