Chains That Look the Same Run Backwards
Last timeHow Long Until It Gets There
Some chains balance pair by pair rather than only situation by situation. When a chain does, its resting distribution can be read straight off the rule with no equations to solve at all.
Finding a resting distribution means solving a linear system. For a chain with
a million situations that is a serious computation, and for a chain whose rule
depends on a parameter you want to vary it has to be redone every time.
A large class of chains escapes this entirely. Their resting distribution can
be written down by looking at the rule. The property that makes it possible is
stronger than resting, and it has a physical meaning: the chain looks the same
run backwards.
A stronger kind of balance
The previous lesson asked that the total flow into each situation match the
total flow out. Nothing in that required any individual pair to match. The
flow could arrive from one neighbour and leave towards another, going round in
a circle, and the totals would still balance.
Detailed balance rules that out. It asks every pair to match on its own.
Adding these up shows that detailed balance is enough. Fix a situation and add
its pair conditions over every partner. The left side becomes its share
multiplied by the total chance of hopping anywhere, which is just its share.
The right side becomes the total arriving from everywhere. So the distribution
is resting.
The reverse is false. A chain on three situations that hops clockwise nine
times in ten has the even split as its resting distribution, and sitting at that
distribution it still carries a steady clockwise circulation. The flow from one
situation to the next is nine times the flow back. Resting, and not reversible.
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