The Ring That Forgets
Last timePushing at the Right Moment
Heat spreading round a ring uses the same shapes as a vibrating string, but they decay instead of oscillating, and the fine detail decays first. What survives is a single number.
A ring of metal is heated at one spot and left alone. Everybody knows what happens
next. What is worth noticing is how little has to change in the machinery of the last
six lessons to say exactly what happens next, and in what order.
- how hot the ring is at the jth point around it. The points are joined in a loop, so the last is a neighbour of the first
- how fast that temperature is changing. One dot, not two: this is a speed of change, not an acceleration of change
- twice the amount by which the average of the two neighbours exceeds this point. Positive when both neighbours are hotter, so the point warms
- how readily heat moves between neighbours. It sets the overall pace and nothing else: it cannot change which shapes decay first
Watching one ring forget
Four points round a loop, all the heat starting in one of them. The quantity held
fixed is the total, and the rule conserves it exactly, because whatever leaves one
point arrives at a neighbour.
the quantity that holdsunder 0.001% of it gone by the end, at every setting of every slider
- stepped by
- x1 → x1 + D*(x2 + x4 - 2*x1) x2 → x2 + D*(x3 + x1 - 2*x2) x3 → x3 + D*(x4 + x2 - 2*x3) x4 → x4 + D*(x1 + x3 - 2*x4)
- each rule reads
- the state as it was when the step began
- started at
- x1 = 1.0, x2 = 0.0, x3 = 0.0, x4 = 0.0
- held steady
- the quantity that holds = x1 + x2 + x3 + x4
- drift
- under 0.001% over the run, against 0.20% allowed
Two symmetries are visible there and both are forced rather than arranged. The two
neighbours behave identically because the starting state and the rule are both
unchanged by reflecting the ring. The end state is a flat quarter each because the
total is conserved and the flat shape is the only one the rule leaves alone.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents