How easy are physical systems to predict?
Well, here are a couple of rather lovely examples. Both of these are due to Michael Berry and are mentioned in a book called ‘A Passion for Science’ which is in fact a set of collected transcripts of BBC radio programmes from sometime in the mid 1980s: I heard them on the radio originally, and they have stayed with me – I didn’t find the paper versions until quite recently. I’m giving these from memory: they might differ slightly from the versions described in the book.
For both of them imagine a universe where everything is completely Newtonian, so no quantum mechanics in particular. There is Newtonian gravity.
The first case is billiards, and we’ll consider a completely idealised billiard table: completely smooth, flat and rigid, completely round balls with completely known properties (so how elastic they are etc), and the same for the cushions. Now someone makes a shot, and we either know the direction and force exactly or are allowed to measure the cue ball’s position and velocity exactly shortly after the shot. We don’t know one thing: there are some people standing around the table, and we don’t know where they are, so we don’t know what their gravitational fields look like. Now we want to predict where the balls go, and we’ll say that the prediction fails when a ball leaves a collision 90 degrees from where we predicted – it’s obvious that after that point we can’t usefully predict anything. How many collisions can we predict ahead?
The electron at the edge of the universe
The second case is an ideal gas: a lot of little ideal particles in an ideal box. Again we know everything: the starting conditions are known completely, the box is completely understood &c &c. The box insulates everything but (Newtonian) gravity to make things simpler. And this time we also know everything about the rest of the universe as well: we don’t need to predict it forward, we’re just given all the data about how it evolves (in fact I think that without loss of generality we can assume an empty universe outside the box, which reduces the data volume considerably). Except that there’s an electron at the edge of the universe and we don’t know where it is (apart from how far away it is), and so again we don’t know its gravitiational field. Now we want to predict this system forwards and we’ll use the same criterion for failure: when some particle leaves a collision 90 degrees out from where we predict. How many collisions before that happens?
The answers are seven or eight for the first case, and about fifty for the second.