Where Nothing Moves
Last timeA State and a Rule
A resting state is one at which every rate is zero at once, so the system left there stays there. Finding them is algebra, not simulation, and it is the first thing worth doing to any rule.
The valley table from the last lesson answers one question at a time and explains
nothing. This lesson asks the first question that can be answered about the rule
itself, with a pencil, without advancing anything.
The question is: is there a state at which this system would not move at all?
- the rule, unchanged from the last lesson. It is the same object being used in a different way, interrogated rather than iterated
- the unknown. A state, meaning a complete list of numbers, and the thing being solved for. The star is a convention for a state that rests
- zero in every component at once. One rate being zero is not enough, and that simultaneity is what makes this a system of equations
Every rate, zero, at the same moment
The word simultaneously is doing real work. Consider a valley where the rabbit count
happens to be steady this year. If the fox count is climbing, next year the rabbit
rate is read from a state with more foxes in it, and it will not be zero. Nothing
rests unless everything rests.
Trying candidates
Before solving, it is worth seeing what being close to a resting state does and does
not look like, because the near misses are instructive.
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