Many Parts, One Total
Last timeRound and Round, and What Holds It There
A state with five numbers instead of two changes nothing about the method. One local rule, plus a total that cannot move, settles where the whole thing ends up before a step is taken.
Every system in this course so far has had a state of two numbers, which has the
pleasant property that it can be drawn. Most real systems are not like that. A metal
bar cooling has a temperature at every point along it, a company has a cash balance
in every account, a city has a crowd density on every street. The state is a list,
and the list can be long.
This lesson makes the point that nothing in the method cares.
One rule, written once
Cut the bar into five sections and track the temperature of each. Heat moves from
hotter to cooler across every join, at a rate set by how big the temperature
difference across that join is. That sentence is the whole rule, and written down it
is this.
- the temperature of section i, the part being updated
- how large a share of a temperature difference crosses a join in one step. Bigger means faster spreading, and too big breaks it, which is the subject of the last lesson of this course
- what the two neighbours have, compared with twice what this section has. It is positive when the section is cooler than its surroundings on average, so the section warms, and negative when it is hotter, so it cools. It is zero when the section sits exactly halfway between its neighbours, which is why the flat state is the one that stops moving
The two end sections are the exception: they have one neighbour instead of two. The
choice made here is that the ends are insulated, so no heat leaves the bar at all.
A different choice, such as holding the ends at the temperature of the room, gives a
completely different answer, and the two look identical until you notice which was
assumed.
The lesson stops here
4 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