Framing Practicum: Dynamic Equilibrium

post by johnswentworth · 2021-08-16T18:52:00.632Z · LW · GW · 2 comments

This is a question post.

Contents

  What’s Dynamic Equilibrium?
  What To Look For
  Useful Questions To Ask
  The Challenge
    If you get stuck, look for:
  Motivation
None
  Answers
    7 AllAmericanBreakfast
    4 adamShimi
    4 Jemist
    3 Elliot Callender
    3 NaiveTortoise
    3 Thoroughly Typed
None
2 comments

This is a framing practicum [? · GW] post. We’ll talk about what dynamic equilibrium is, how to recognize dynamic equilibrium in the wild, and what questions to ask when you find it. Then, we’ll have a challenge to apply the idea.

Today’s challenge: come up with 3 examples of dynamic equilibrium which do not resemble any you’ve seen before. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you). 

Expected time: ~15-30 minutes at most, including the Bonus Exercise.

What’s Dynamic Equilibrium?

Our picture of stable equilibrium [? · GW] was a marble sitting at the bottom of a bowl. This is a static equilibrium: the system’s state doesn’t change.

Picture instead a box full of air molecules, all bouncing around all over the place. The system’s state constantly changes as the molecules move around. But if we look at the box at large scale, we don’t really care about the motions of individual molecules, we just care about the overall distribution of molecule positions and velocities - the number of molecules in the upper-left quadrant, for instance. And this distribution may be (approximately) constant, even though the individual molecules are bouncing around.

This is dynamic equilibrium: even though the states of the system’s components are constantly changing, the distribution of states of the system’s components has a stable equilibrium.

The box-of-air example involves a distribution of similar physical parts (i.e. molecules), but we can also have a dynamic equilibrium with a Bayesian belief-distribution. For instance, I can think about which of my dishes I expect to be dirty tomorrow or next week or next month. I don’t run the dishwasher every day, so in the short term I expect dirty dishes to pile up - a non-equilibrium expectation. But in the long run, I generally expect the distribution of dirty dishes to be roughly steady - I don’t know exactly which days I’ll wash them, but my expectations for 100 days from now are basically the same as my expectations for 101 days from now.

The dirty dishes themselves may pile up and then be cleaned and then pile up again, never reaching an equilibrium state. But my expectations or forecasts about the dishes do reach an equilibrium.

What To Look For

In general, dynamic equilibrium should spring to mind in two situations:

• A system has parts which are constantly changing/moving, but some aggregate summary of those parts (like a total count or a distribution) tends to return to approximately the same value.
• Our expectations or forecasts about a system’s state tend to return to approximately the same thing when we forecast farther into the future.

Useful Questions To Ask

If the number of air molecules in the upper-left quadrant of a box is roughly constant, then the rate at which molecules enter that quadrant must roughly equal the rate at which they leave. If the number of molecules in the quadrant is lower than usual, then molecules will enter faster than they leave until the number returns to equilibrium.

In general, dynamic equilibrium involves a balance: the rate at which parts enter some state is roughly equal to the rate at which they leave that state. Three key questions are:

• At what rate do parts enter each state?
• At what rate do parts leave each state?
• What does the state-distribution have to look like for those to balance?

In order for the equilibrium to be stable, we also need parts to enter a state a bit faster/leave it a bit slower when there are fewer-than-equilibrium-number of parts in that state. So, besides the rates, we also want some idea of how the rates change if there are slightly more or less parts in a given state.

We can also ask the corresponding questions for a dynamic equilibrium of expectations/forecasts. Rather than parts moving between states, we have probability-mass moving between states. If there’s a chance that I wash the dishes in two days, then there’s a flow of probability-mass from the “n dirty dishes” state to the “all dishes clean state” between two and three days in the future. If my expectations reach an equilibrium, then that means the rate of probability-mass-flow into each state equals the rate of probability-mass-flow out. So, the three key questions are:

• What processes cause probability-mass to flow into each state, and at what rate do those happen?
• What processes cause probability-mass to flow out of each state, and at what rate do those happen?
• What does the distribution have to look like for those to balance?

Once we know how the state-change rates work, we can also ask all the usual questions about stable equilibrium [? · GW].

The Challenge

Come up with 3 examples of dynamic equilibrium which do not resemble any you’ve seen before. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you).

Any answer must include at least 3 to count, and they must be novel to you. That’s the challenge. We’re here to challenge ourselves, not just review examples we already know.

However, they don’t have to be very good answers or even correct answers. Posting wrong things on the internet is scary, but a very fast way to learn, and I will enforce a high bar for kindness in response-comments. I will personally default to upvoting every complete answer, even if parts of it are wrong, and I encourage others to do the same.

Post your answers inside of spoiler tags. (How do I do that? [LW · GW])

Celebrate others’ answers. This is really important, especially for tougher questions. Sharing exercises in public is a scary experience. I don’t want people to leave this having back-chained the experience “If I go outside my comfort zone, people will look down on me”. So be generous with those upvotes. I certainly will be.

If you comment on someone else’s answers, focus on making exciting, novel ideas work — instead of tearing apart worse ideas. Yes, And is encouraged. 

I will remove comments which I deem insufficiently kind, even if I believe they are valuable comments. I want people to feel encouraged to try and fail here, and that means enforcing nicer norms than usual.

If you get stuck, look for:

• Systems made of lots of similar parts which are all constantly in motion
• Systems with a stable equilibrium only if you “zoom out” from the details
• Systems for which your expectations are roughly constant sufficiently far into the future, even if the system itself is constantly in motion.

Bonus Exercise: for each of your three examples from the challenge, what are the relevant “parts”, part-states and state-change rates (or probability-mass-flows, for expectations)? Can you do a Fermi estimate of the relevant rates, or estimate a rough (i.e. big-O) functional relationship between the state-change rates and the number of parts in each state (or probability-mass in each state)? How do the state-change rates change when the number of parts (or probability mass) in some state is higher/lower than its equilibrium value?

This bonus exercise is somewhat more abstract and conceptually tricky than previous exercises, especially for the probability-mass questions. I recommend it especially if you want some extra challenge.

Motivation

Much of the value I get from math is not from detailed calculations or elaborate models, but rather from framing tools: tools which suggest useful questions to ask, approximations to make, what to pay attention to and what to ignore.

Using a framing tool is sort of like using a trigger-action pattern [? · GW]: the hard part is to notice a pattern, a place where a particular tool can apply (the “trigger”). Once we notice the pattern, it suggests certain questions or approximations (the “action”). This challenge is meant to train the trigger-step: we look for novel examples to ingrain the abstract trigger pattern (separate from examples/contexts we already know).

The Bonus Exercise is meant to train the action-step: apply whatever questions/approximations the frame suggests, in order to build the reflex of applying them when we notice dynamic equilibrium.

Hopefully, this will make it easier to notice when a dynamic equilibrium frame can be applied to a new problem you don’t understand [LW · GW] in the wild, and to actually use it.

Answers

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