## The End 2016 Mathematics A To Z: Ergodic

This essay follows up on distributions, mentioned back on Wednesday. This is only one of the ideas which distributions serve. Do you have a word you’d like to request? I figure to close ‘F’ on Saturday afternoon, and ‘G’ is already taken. But give me a request for a free letter soon and I may be able to work it in.

## Ergodic.

There comes a time a physics major, or a mathematics major paying attention to one of the field’s best non-finance customers, first works on a statistical mechanics problem. Instead of keeping track of the positions and momentums of one or two or four particles she’s given the task of tracking millions of particles. It’s listed as a distribution of all the possible values they can have. But she still knows what it really is. And she looks at how to describe the way this distribution changes in time. If she’s the slightest bit like me, or anyone I knew, she freezes up this. Calculate the development of millions of particles? Impossible! She tries working out what happens to just one, instead, and hopes that gives some useful results.

And then it does.

It’s a bit much to call this luck. But it is because the student starts off with some simple problems. Particles of gas in a strong box, typically. They don’t interact chemically. Maybe they bounce off each other, but she’s never asked about that. She’s asked about how they bounce off the walls. She can find the relationship between the volume of the box and the internal gas pressure on the interior and the temperature of the gas. And it comes out right.

She goes on to some other problems and it suddenly fails. Eventually she re-reads the descriptions of how to do this sort of problem. And she does them again and again and it doesn’t feel useful. With luck there’s a moment, possibly while showering, that the universe suddenly changes. And the next time the problem works out. She’s working on distributions instead of toy little single-particle problems.

But the problem remains: why did it ever work, even for that toy little problem?

It’s because some systems of things are ergodic. It’s a property that some physics (or mathematics) problems have. Not all. It’s a bit hard to describe clearly. Part of what motivated me to take this topic is that I want to see if I can explain it clearly.

Every part of some system has a set of possible values it might have. A particle of gas can be in any spot inside the box holding it. A person could be in any of the buildings of her city. A pool ball could be travelling in any direction on the pool table. Sometimes that will change. Gas particles move. People go to the store. Pool balls bounce off the edges of the table.

These values will have some kind of distribution. Look at where the gas particle is now. And a second from now. And a second after that. And so on, to the limits of human knowledge. Or to when the box breaks open. Maybe the particle will be more often in some areas than in others. Maybe it won’t. Doesn’t matter. It has some distribution. Over time we can say how often we expect to find the gas particle in each of its possible places.

The same with whatever our system is. People in buildings. Balls on pool tables. Whatever.

Now instead of looking at one particle (person, ball, whatever) we have a lot of them. Millions of particle in the box. Tens of thousands of people in the city. A pool table that somehow supports ten thousand balls. Imagine they’re all settled to wherever they happen to be.

So where are they? The gas particle one is easy to imagine. At least for a mathematics major. If you’re stuck on it I’m sorry. I didn’t know. I’ve thought about boxes full of gas particles for decades now and it’s hard to remember that isn’t normal. Let me know if you’re stuck, and where you are. I’d like to know where the conceptual traps are.

But back to the gas particles in a box. Some fraction of them are in each possible place in the box. There’s a distribution here of how likely you are to find a particle in each spot.

How does that distribution, the one you get from lots of particles at once, compare to the first, the one you got from one particle given plenty of time? If they agree the system is ergodic. And that’s why my hypothetical physics major got the right answers from the wrong work. (If you are about to write me to complain I’m leaving out important qualifiers let me say I know. Please pretend those qualifiers are in place. If you don’t see what someone might complain about thank you, but it wouldn’t hurt to think of something I might be leaving out here. Try taking a shower.)

The person in a building is almost certainly not an ergodic system. There’s buildings any one person will never ever go into, however possible it might be. But nearly all buildings have some people who will go into them. The one-person-with-time distribution won’t be the same as the many-people-at-once distribution. Maybe there’s a way to qualify things so that it becomes ergodic. I doubt it.

The pool table, now, that’s trickier to say. For a real pool table no, of course not. An actual ball on an actual table rolls to a stop pretty soon, either from the table felt’s friction or because it drops into a pocket. Tens of thousands of balls would form an immobile heap on the table that would be pretty funny to see, now that I think of it. Well, maybe those are the same. But they’re a pretty boring same.

Anyway when we talk about “pool tables” in this context we don’t mean anything so sordid as something a person could play pool on. We mean something where the table surface hasn’t any friction. That makes the physics easier to model. It also makes the game unplayable, which leaves the mathematical physicist strangely unmoved. In this context anyway. We also mean a pool table that hasn’t got any pockets. This makes the game even more unplayable, but the physics even easier. (It makes it, really, like a gas particle in a box. Only without that difficult third dimension to deal with.)

And that makes it clear. The one ball on a frictionless, pocketless table bouncing around forever maybe we can imagine. A huge number of balls on that frictionless, pocketless table? Possibly trouble. As long as we’re doing imaginary impossible unplayable pool we could pretend the balls don’t collide with each other. Then the distributions of what ways the balls are moving could be equal. If they do bounce off each other, or if they get so numerous they can’t squeeze past one another, well, that’s different.

An ergodic system lets you do this neat, useful trick. You can look at a single example for a long time. Or you can look at a lot of examples at one time. And they’ll agree in their typical behavior. If one is easier to study than the other, good! Use the one that you can work with. Mathematicians like to do this sort of swapping between equivalent problems a lot.

The problem is it’s hard to find ergodic systems. We may have a lot of things that look ergodic, that feel like they should be ergodic. But proved ergodic, with a logic that we can’t shake? That’s harder to do. Often in practice we will include a note up top that we are assuming the system to be ergodic. With that “ergodic hypothesis” in mind we carry on with our work. It gives us a handle on a lot of problems that otherwise would be beyond us.

## Reading the Comics, September 6, 2015: September 6, 2015 Edition

Well, we had another of those days where Comic Strip Master Command ordered everybody to do mathematics jokes. I’ll survive.

Don Trachte’s Henry for the 6th of September, 2015.

Don Trachte’s Henry is a reminder that arithmetic, like so many things, is easier to learn when you’re comfortable with the context. Personally I’ve never understood why some of the discs on pool scoring racks are different colors but imagine it relates to scoring values, somehow. I’ve encountered multiple people who assume I must be good at pool, since it’s all geometry, and what isn’t just geometry is physics. I’ve disappointed them all so far.

Tony Rubino and Gary Markstein’s Daddy’s Home uses arithmetic as an example of joy-crushing school drudgery. It could’ve as easily been asking the capital of Montana.

Scott Adams’s Dilbert Classics, a rerun from the 29th of June, 1992, has Dilbert make a breakthrough in knot theory. The fundamental principle is correct: there are many knots that one could use for tying shoelaces, just as there are many knots that could be used for tying ties. Discovering new ones is a good ways for knot theorists to get a bit of harmless publicity. Nobody needs them. From a knot-theory perspetive it also doesn’t matter if you lace the shoe’s holes crosswise or ladder-style. There are surely other ways to lace the holes, too, but nobody needs them either.

Maria Scrivan’s Half Full uses a blackboard full of mathematical symbols and name-drops Common Core. Fifty years ago this same joke was published, somewhere, with “Now solve it using the New Math” as punchline. Thirty years from now it will run again, with “Now solve it using the (insert name here)” as punchline. Some things are eternal truths.

T Lewis and Michael Fry’s Over The Hedge presents one of those Cretan paradox-style logic problems. Anyway, I choose to read it as such. I’m tickled by it.

And to close things out, both Leigh Rubin’s Rubes and Mikael Wulff and Anders Morgenthaler’s WuMo did riffs on the story of Newton and the falling apple. Is this truly mathematically-themed? Well, it’s tied to the legend of calculus’s origin, so that’s near enough for me.

• #### ivasallay 3:48 pm on Tuesday, 8 September, 2015 Permalink | Reply

Speaking of knot theory and tying shoes, the third thing that Matt Parker does in this YouTube video is show the most mathematically precise way to tie your shoes: https://www.youtube.com/watch?v=1wAaI_6b9JE

Like

• #### Joseph Nebus 11:46 pm on Saturday, 12 September, 2015 Permalink | Reply

That is a really neat-looking video. I’m going to have to watch it as soon as I can get to a place with better bandwidth. (Our home Internet is terrible for videos, unfortunately, especially as YouTube doesn’t let you buffer worth anything anymore.)

Like

• #### Joe Nebus 4:55 pm on Tuesday, 8 September, 2015 Permalink | Reply

Henry was a fun strip I followed as a kid… thanks for the memory, signed
Old Geezer

Like

• #### Joseph Nebus 11:43 pm on Saturday, 12 September, 2015 Permalink | Reply

Glad you like. The comic strip is still running, although it seems to be all reruns. (I guess that’s part of the aesthetic of the Henry comic.) I can’t pin down when they’re from, though, or who did the daily strips that’re repeating.

Like

## Reblog: Making Your Balls Bounce

Neil Brown’s “The Sinepost” blog here talks about an application of mathematics I’ve long found interesting but never really studied, that of how to simulate physics for game purposes. This particular entry is about the collision of balls, as in for a billiard ball simulation.

It’s an interesting read and I do want to be sure I don’t lose it.

Like

In this post, we will finally complete our pool game. We’ve already seen how to detect collisions between balls: we just need to check if two circles are overlapping. We’ve also seen how to resolve a collision when bouncing a ball off a wall (i.e. one moving object and one stationary). The final piece of the puzzle is just to put it all together in the case of two moving balls.

## Bouncy Balls

The principle behind collision resolution for pool balls is as follows. You have a situation where two balls are colliding, and you know their velocities (step 1 in the diagram below). You separate out each ball’s velocity (the solid blue and green arrows in step 1, below) into two perpendicular components: the component heading towards the other ball (the dotted blue and green arrows in step 2) and the component that is perpendicular to the other…

View original post 303 more words

c
Compose new post
j
Next post/Next comment
k
Previous post/Previous comment
r