In working out my little Arthur Christmas-inspired problem, I argued that if the reindeer take some nice rational number of hours to complete one orbit of the Earth, eventually they’ll meet back up with Arthur and Grand-Santa stranded on the ground. And if the reindeer take an irrational number of hours to make one orbit, they’ll never meet again, although if they wait long enough, they’ll get pretty close together, eventually.
So far this doesn’t sound like a really thrilling result: the two parties, moving on their own paths, either meet again, or they don’t. Doesn’t sound quite like I earned the four-figure income I got from mathematics work last year. But here’s where I get to be worth it: if the reindeer and Arthur don’t meet up again, but I can accept their being very near one another, then they will get as close as I like. I only figured how long it would take for the two to get about 23 centimeters apart, but if I wanted, I could wait for them to be two centimeters apart, or two millimeters, or two angstroms if I wanted. I’d pay for this nearer miss with a longer wait. And this gives me my opening to a really stunning bit of mathematics.
To get there, I need to introduce the idea of a “phase space”. This is part of the grand mathematics project of turning things into pictures because pictures are much easier to understand than numbers and equations are. A phase space is a picture where the different coordinates are whatever properties define the problem you’re looking at. The example math students learn on is the phase space for a pendulum bob, because there are two coordinates that matter: the angle the pendulum makes with respect to the vertical, and the speed with which the pendulum is swinging.
What’s great about phase space diagrams is that you can look at how a system develops as a picture. The changing of the positions and momentums of particles in it are going to appear as a path traced out within that phase space, just like the ground track of the reindeer and sleigh would be. The picture can explain much about how a system works, letting you understand what is going to happen, before you ever work out any particular solutions to the problem.
For example, with the swinging pendulum, you’ll see the phase space divided into the range where the swing simply rocks back and forth and the range where there’s so much speed that it keeps spinning around over and over, like Wile E Coyote were swinging with a rocket pack strapped on. This works even if you have confounding factors like friction in the swing: you can divide phase space into “the swing rocks back and forth, ultimately stopping” and “the swing spins around once and then rocks back and forth” and “the swing spins around twice and then rocks back and forth”, “the swing spins around three times and then rocks back and forth”, and so on.
Any physical system, as it develops in time, is going to trace out a path within phase space. This is usually called a trajectory or an orbit, even though it may not look much like the orbits of planets or satellites which the word immediately evokes. An orbit is either going to have to make a closed loop — it’ll come back around to whatever the state was when the system started — or else it won’t. Again, not the most stunning of results: the path is either a loop or it isn’t.
Now, there’s a lot of tidying up and tightening of logical loopholes to go from the Arthur-and-reindeer problem to the development of a physical system in phase space. However, if you imagine Arthur and Grand-Santa, staying where they landed, as representing the original state of the system, and the reindeer flying on as the development of the system in time, then you can get an idea of what the evolution of a physical system in time is like, using ideas that are pretty easy to visualize. This is why I am only a little ashamed of making this leap, and of asking you to make it with me.
With the paths, that either repeat themselves or don’t, along comes the Poincaré Recurrence Theorem. This was published in 1890 by Henri Poincaré, a name which immediately leaps to mind as that of the President of the French Republic during World War I. Wrong Poincaré. You’re thinking of Raymond Poincaré, Henri’s cousin. Henri Poincaré is one of those names that comes up in all kinds of problems about dynamics and geometry and probability in the late 19th and early 20th century. He’s also usually named the person who probably would’ve discovered General Relativity had Einstein not got there first. He’s also got a decent biographical anecdote about a baker whose breads, Poincaré believed, were consistently underweight that’s good if you need biographical anecdotes about Henri Poincaré for some reason. (I offer no opinion about whether it’s true.)
According to the Recurrence Theorem, though, if a physical system follows laws obeying the conservation of energy, and some other technical points, then, it is either periodic — it will get back to the starting state and repeat itself perfectly — or else it will, given enough time, come arbitrarily close to the starting state. Just as with reindeer, they may not get exactly back to where Arthur and Grand-Santa, representing the starting point, left off, but they’ll come as close as you like if you wait long enough.
This gets exciting when you remember that all physical interactions conserve energy. Even the ones that we normally write off, like friction or air resistance or wasted heat, really are conserving energy and momentum and all the other conservation laws. Normally, we don’t care about the air stirred up by the swing going back and forth, so we don’t keep track of that, and pretend that there’s energy lost to it; but it isn’t, not really. If we kept track of all the particles in the universe, we’d see, the total energy is conserved through all the interactions they have.
The implication then is that if you waited long enough, all the particles in the universe would eventually get back to the same positions and momentums as … oh, let’s say, now. Or if they didn’t get exactly back, they’d at least get close enough. If we got the entire universe re-created in every detail except there’s a pebble on Mars that’s out of place you wouldn’t care, would you?
As with any sort of pronouncements about the long-term fate of the universe this is a bit explosive. The explosion came pretty quickly, in statistical mechanics circles, in the form of a good solid fight between Ernst Zermelo, a brilliant man who put set theory on a solid basis, and Ludwig Boltzmann, a brilliant man who put statistical mechanics on a solid basis. Zermelo pointed out that, since entropy is a physical quantity, something that can be described in terms of the positions and momentums and interactions of particles in the universe, and since entropy has to increase — second law of thermodynamics, remember, and put on a rigorous basis by something called Boltzmann’s H-Theorem — then the conclusion that the universe could get back to where it started meant its entropy might increase but it must then decrease, and that’s absurd.
Boltzmann argued back that the H-Theorem didn’t say entropy always increased; just that, it’s much more likely to increase than to decrease, in the same way that shuffling a deck of cards is much more likely to leave you with cards in a scrambled order than cards put back in order. But it’s not impossible that shuffling will put the cards back in order; you just need to keep trying long, long, long enough.
Along the way to this Boltzmann worked out approximately how long it would take a mere one trillion molecules of air, in a volume one cubic centimeter in size, to get to the first recurrence. The number of seconds this takes is a number with “many trillions of digits” in it. Considering that a googol of seconds is a touch under 320, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000, 000,000 years — and that’s a number of seconds only a hundred digits long — it seems fair to say the times involved allow, first, for many improbable things to happen; and second, that even that’s apparently not long enough for the whole universe to recur.
Of course, if you have infinitely long to wait … well, could the universe repeat even after unspeakably many googolplexes of years? Either way is a staggering and an unsettling thought.
It’s also one I’m surprised more science fiction writers haven’t used. Looping or repeating universes seem to be accepted — there was one pretty interesting issue of the Gold Key Star Trek comic book with it (according to the comic’s “Free Will Theorem” there’s one person in all time and space who can make a free decision and break the chain of endless repeating universes and, you guessed it, it’s none other than NCC-1701’s very own Lieutenant DeSalle) and Futurama did a disappointing episode with the gimmick — but ones where the recurrence is actually scientifically justified seem rare. Perhaps they suppose the aesthetic reasons to do a repeating universe are enough they don’t need science. I’m also surprised that it hasn’t been taken as proof of reincarnation by people who want to grab slender straws of science to hold up their belief in the paranormal. Since in this construction everyone would just be reincarnated as themselves living the lives they already do already perhaps that takes the fun out of it.
As with any proclamation about the extremely long-term fate of the universe, the answer is unsure. And no, the trouble isn’t that there are quantum mechanics things producing weirdness in the universe; the recurrence theorem can be written in a nice quantum-friendly fashion. But it’s not obvious that all of the prerequisites for the recurrence theorem to apply actually are met, in the universe as it actually is, instead of as a nicer mathematical model suggests.
That’s about as much as I can say about Arthur Christmas, at least for now, though I can’t promise I won’t return to the subject, for obvious reasons.