Nudge It, and See Whether It Comes Back
Last timeWhere Nothing Moves
Displace a system slightly from a resting state and the rule either shrinks the displacement or grows it. The deciding quantity is the slope of the rate, and its sign is the answer.
A marble at the bottom of a bowl and a pencil balanced on its point are the same as
far as the previous lesson is concerned. Both are states where nothing moves. One of
them you will actually find in the world and the other you will not.
The difference has nothing to do with the state itself and everything to do with the
neighbourhood, so this lesson looks slightly to one side of the resting state.
What happens to a small displacement
Put the system a small distance from a resting state and follow the displacement
rather than the state. The rule evaluated at the resting state gives zero, by
definition. The rule evaluated a little away from it gives approximately the slope of
the rule there times how far away we are, which is the straight-line approximation
from the previous course used in anger for the first time.
- how far the system is from the resting state after k steps. Small, and assumed to stay small, which is the assumption the whole lesson rests on
- the slope of the rate curve at the resting state. One number, computed once, and the only thing about the rule that survives into this approximation
- the step of time, as before. It scales how much happens per step and cannot change the sign of anything
- the factor the displacement is multiplied by each step. Smaller than one when the slope is negative, so the displacement shrinks; larger than one when the slope is positive, so it grows
So the question of whether a system returns to a resting state has collapsed into the
sign of a single number. Negative slope, displacement shrinks, the system comes back.
Positive slope, displacement grows, the system leaves.
The lesson stops here
5 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