[ Question ]

Framing Practicum: Stable Equilibrium

by johnswentworth5 min read9th Aug 202130 comments



This is a framing practicum post. We’ll talk about what a stable equilibrium is, how to recognize stable equilibria in the wild, and what questions to ask when you find one. Then, we’ll have a challenge to apply the idea.

Today’s challenge: come up with 3 examples of stable 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 a Stable Equilibrium?

Put a marble at the bottom of a round bowl, and it will just sit there without moving. Put it in the bowl but not quite at the bottom, and it will roll around a bit, but eventually settle at the bottom, and sit there without moving. Give it a poke, and it will roll around some more, but eventually it will again sit at the bottom without moving.

This is stable equilibrium: the system may start in different states, or it may be “perturbed” into different states by some external force, but eventually it settles back to the same state (assuming it isn’t pushed too far away…).

A marble in a bowl will eventually sit stationary at the bottom of the bowl, and stay there.

What To Look For

Stable equilibrium should spring to mind whenever a system tends to return to the same state. If you could “poke” it somehow, and the system would go back to normal eventually, that’s probably a stable equilibrium. If a system tends to stay suspiciously the same over the long run, despite lots of short-run noise, that’s probably a stable equilibrium.

Useful Questions To Ask

The marble always returns to the bottom of the bowl. If we push the marble away from the bottom, that’s only a short-term change - it will roll back down eventually. So, if we’re mainly interested in the long run behavior of the marble, then we can ignore such little pushes.

On the other hand, there may also be ways to change the equilibrium state itself. For instance, if we tip the bowl to the side slightly, then the equilibrium position of the marble will change. If we deform the bowl, that could change equilibrium position. If we charge the marble with a little static electricity, then place another charged object near the bowl, that could also change the equilibrium. Finally, very large changes to the system state could push it out of the bowl entirely.

Tipping the bowl changes the equilibrium.

When we frame something as a stable equilibrium, we ignore temporary changes to the system state, and only pay attention to things which change the equilibrium.

Two main ways this can apply:

  • We want to change the long-term behavior of a system. So, we focus on things which can change the equilibrium, and ignore things which don’t.
  • We see a change in the long-term behavior of a system, and want to know what caused it. So, we focus on things which can change the equilibrium, and ignore things which don’t.

The Challenge

(Rules adapted from the Babble Challenges)

Come up with 3 examples of stable 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). I recommend mentioning what the equilibrium is, and a few ways you could “poke” the system for which it would return to equilibrium afterwards, so that everyone understands your example.

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?)

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.

Reward people for babbling — don’t punish them for not pruning. 

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 which go back to normal after you poke them
  • Systems which stay suspiciously the same over time despite lots of short-term noise

Bonus Exercise: for each of your three examples from the challenge, suppose you want to change the equilibrium, or you want to know what caused a change in the equilibrium. What factors should you pay attention to (since they can change the equilibrium)? What factors can you safely ignore (since they only affect the system in the short term)?

This bonus exercise is great blog-post fodder!


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: 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 a stable equilibrium.

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

Thankyou to Sisi, Eli, Adam and especially Jacob for beta-testing and feedback. Also thankyou to Aysajan for our daily discussions, which led to this concept.


New Answer
Ask Related Question
New Comment

13 Answers

  1. The number of houses on fire in a metropolitan area. It's generally around zero. Whenever the number of fires goes up, firemen show up and reduce the number until it's close to zero again. Separately, if there is a lot of fire, a lot of things might burn down, but then the fire burns itself out. 
  2. A very heavy rock. Because it's heavy, it doesn't tend to change.
  3. Salt levels in human blood. If it gets too low, human eats more salt. If it gets too high, liver gets rid of it.

The second one is interesting to me because if you increase weight by caking it in mud, the mud will break/fall/rub off, and the rock will return to its previous weight. But if you break off a piece, it will generally not return to its previous weight. Maybe a version of this that returns to equilibrium from both directions is a car? If you break a reasonable number of pieces off or put wear on the tires or burn some gas or oil, it will return to its 'equilibrium' weight via maintenance?

I particularly like the first one. Highly relevant to current events in California, at multiple timescales.

  1. The even surface of a pond. If I throw in a stone it'll ripple for a while, but the additional activity will dissipate again.
  2. The number of tabs I have open in my browser. If it's very few I have no problem opening and keeping new ones. If it's too many, titles become hard to read etc. and I feel reluctant to keep open additional ones and tend to close a bunch of them at once.
  3. The amount of social interaction I get. Too little and I feel lonely and reach out to people. Too much and I feel overwhelmed and keep to myself for a while.

I sympathize with #2, although for me the "titles become hard to read" issue is a lost cause - my equilibrium is way past that.

  1. The dog at the Burrow has a suspiciously regular patrol, during which he looks for food. If you give him food, he will lick your hand and wag his tail gratefully, before returning to the patrol. 

    If someone took him away in a car, he would not return to the route.
  2. When I move my mouse, my monitor activates. If I do nothing for a while, it will go dark again. 

    If I set up a robot which constantly twitched the mouse, the monitor would not go dark, no matter how long I waited.
  3. An elastic object, such as a rubber band, will hang such that the force of tension equals the force of gravity. 

    If I snap the band, it will not return to its original equilibrium.

The first one highlights a useful sub-frame of stable equilibrium: stable cycles (aka limit cycles). The equilibrium isn't just one state, it's a cycle of states. Perturb it, and eventually it goes back to the same cycle.

My answers from when John asked me to try the exercise a couple of days ago:

  • A message with error-correcting codes
  • A room heated with a thermostat
  • Water in a glass
  • Profound distaste for a specific food. Whatever the small perturbation in taste made to it through cooking, you still find it disgusting (for example one of my brothers hate potatoes)
  • Most people's political belief, where small perturbations (contradiction, a favorite politician condemned for fraud, a good policy by the outgroup) aren't enough to change the political belief.
  • Two system monopoly on election: moving a small number of votes from one or both to other small parties doesn't change the duopoly.

And the bonus exercise:

  • A message with error-correcting codes: if I want to change the message or see that it is change, it's important to look for the maximal amount of errors that the code can handle, as well as what are the closest message that result from adding just too much error.
  • A room heated with a thermostat: If the change is lower than the thermostat's temperature, I must look for some way to cool the room faster than the thermostat can heat it, probably with an exchange with the outside and some sort of heat pump. If the change is higher than the thermostat, then I must look for a source of additional heat that never stops (as the thermostat never makes the room cooler, just wait for it too cool down).
  • Water in a glass: some factors are the angle of the glass, its structural integrity, the gravitational force on the water, and whether someone around is thirsty
  • Profound distaste for a specific food: factors seem mainly two, either a change of taste due to age (now I like mushroom but I hated them as a kid), or such a transformation of the food that it loses its problematic properties (like my brother can eat thin fries because they barely taste of potatoes)
  • Most people's political belief: factors like a change of community, a complete change in the community, so many scandals that the person changed their mind, or someone capitalizing on the discontent while not changing the image of the political belief.
  • Two system monopoly on election: factors could be most people agreeing to vote to a third party to break the duopoly, or so few people voting that even small parties have a shot, or siphoning votes for the duopoly into many different parties, splitting everything up.

I particularly like the last two examples. The second-to-last in particular might make an interesting frame for how-to-change-one's-mind.

Main exercise:

  1. Amount of muscle a person who doesn't exercise regularly has.
  2. Level of clutter on a person's desk/counter/etc.
  3. Quantity of light that reaches a forest floor at a given time.
  4. (Extra:) Number of organisms in an all-male group.

I recognize 1 and 3 are borderline dynamic equilibria but I think they changes on a slow enough timescale that they count.

Bonus exercise:

  1. Watch their diet over the timescale of weeks to months and their physical activity. Can ignore incidental activity, like running to catch a bus or lifting lots of stuff for a move. More generally, can ignore any activity that's acute.
  2. Persistent changes to their cleaning behaviors (for reduction) or accumulation patterns (do they switch to computer notetaking?). Can ignore temporary changes or routine cleaning behaviors that have been going on for a while.
  3. Pay attention to introduction of organisms/factors that change the forest to a different type of environment with less (or more) coverage. Can ignore both temporary disturbances like a human walking through and crushing plants and introduction of organisms which only consume one of several types of trees that form the canopy, since others will presumably fill the void.

I particularly like 2 & 3 - they evoke great visualizations in my head. I imagine a fast-forwarded video showing things appearing and disappearing, but density staying at roughly the same level over time.

A fire burning out. Pokes: adding fuel or oxygen.

Solving a puzzle, sorting a list. Pokes: breaking up the puzzle, adding out-of-order elements to the list.

Adoption. Pokes: CPS.

Finding a quality/trustworthy brand. Pokes: degradation in quality, changing preferences.

This exercise became much easier once I shifted my mindset. At first, I was picking a theme, with all the relevant details, then trying to find some way of justifying it as a stable equilibrium. What about being hungry? You start with a plate full of food. Then you eat it. But what if you don't eat all the food? And then you wash the plate, and you might fill it full of food tomorrow. But then again, it feels sort of like once you're full and not eating anymore, you're sort of "stable" for that meal. But then again....

What helped was to realize that I was in charge of determining what is "inside" and "outside" the system. Instead of somehow arguing about what "counts as stable" or trying to pare down details, I could instead choose to define the system by building it up from its simplest elements, only including elements that I wanted to be relevant. By defining the activities of food preparation and digestion as "outside" the system and existing food as "inside" the system, eating lunch arrives at a stable equilibrium where plates are empty and bellies are full.

By changing what counts as "outside" and "inside," we can get a different equilibrium. Our purposes determine which model is more useful. If I am thinking about running a restaurant, then I try to keep a stock of food always available in a dynamic equilibrium. If I am thinking about eating a picnic, then I don't want to carry back a bunch of half-eaten food and am aiming for the stable equilibrium where it's all been eaten.

The resulting model is relatively low-fidelity, but exhibits the properties of interest, is easy to think about, and can always be complexified if necessary. I think it's probably a useful mental habit to cultivate. Instead of working with the most complex model you can manage to manipulate, use the simplest model that can produce useful results.

Good insights. The inside/outside assignment becomes especially important when we have have multiple processes which equilibrate at different timescales - e.g. a commodity price may have both a short-term equilibrium (which just balances near-term supply and demand) and a long-term equilibrium (in which new buyers/sellers start businesses/shut down businesses in response to prices). In that situation, we explicitly declare the long-term changes to be "outside" (aka "exogenous") when analyzing the short-term equilibrium.

It is harder than expected not to recycle from known instances. I had to totally avoid physics and markets to feel like finding not remembering examples.

  •  There's a lot of somewhat cyclic stochastic processes that I would call a stable equilibrium. My whiteboard tends to have about the same fraction of free space most of the time. Sketching something or deleting something is a fluctuation. Changing my usage habits could make a long term difference.
  • The density of social events in my calendar is surprisingly constant. Less is boring, more is exhausting. It regulates itself. Fluctuations include random clusters of important events or little spare time. To constantly change it I could become a recluse or more sociable.
  • The mass of the contents of most rooms is dominated by a few long lasting things and is mostly stable. Getting a cup of tea is a small disturbance. Buying new furniture will suddenly change the equilibrium. 

One nice thing about this exercise is that it gets harder the more you've used the concept in specific contexts (like physics or economics). Well done.

  1. The location of students in a classroom. It's been a few years since I sat in a classroom regularly, but I remember people sitting in the same seats each class, sometimes exactly, sometimes coarsely (e.g. friends sit roughly the same set of seats, with a mostly-random permutation among the group of friends). Perturbations like a one-time guest sitting in someone's seat or a chair being broken for a week will disrupt the seating arrangement, but people will return to their old seats if it doesn't last too long.

  2. Sleeping patterns. I tend to sync up with sunrise or my work schedule, but if I stay up late or wake up early or sleep poorly or something, my sleep will out of sync and eventually find its way back to where it was before.

  3. The messiness of my apartment. Sometimes I'll put in a lot of work to make it very clean and sometimes it will get very messy for some reason, but it tends to return to a relatively stable level of a little messy. Notably, the equilibrium for this has steadily shifted toward less messy as I get older.

Bonus exercise:

  1. My guess is that this is a combination of actual preferences for particular seats (close to the front vs close to the back vs close to the door, for example), a clustering effect from people wanting to sit near friends, and a desire for stability, predictability, and not taking someone else's seat. Changing which seats are desirable according to various criteria seems hard, but you might be able to overcome the desire for stability by rewarding students for sitting with different people or in different parts of the classroom for several lectures, then allowing them to do whatever they want.

  2. Now that I'm thinking about it, this is a big topic for people, but I've had luck with shifting it using melatonin and changing my evening/evening lighting.

  3. I think the equilibrium point lives where the marginal (perceived) effort of cleaning is equal to the marginal (perceived) benefit of having things tidier, minus the marginal (perceived) cost of having everything put away where I can't find it. One possibility is to change my perceptions, though I'm not sure how to do this. Another is to reduce the cost of cleaning or grabbing something that's not already sitting out in front of me, and I think having better organization can help with both of these.

All strong examples. Reading the bonus exercise answers, every single one sounded like an interesting model with some nontrivial insights.

If you flip the bowl over, and set a marble on top, this may be an unstable equilibrium - assuming you can even manage to get it stay there in the first place (equilibrium).

1. A post without any comments on it, may be a stable equilibrium (with regard to total length).

2. Multiple crabs in a bucket.

3. If people pick up trash when there's not a lot of trash, but not if there's a lot, then both 'clean' and 'trash everywhere' can be stable equilibriums.

4. Walking up stairs (or going up an elevator). At a higher level, you maintain your altitude. (And jumping up and down doesn't perturb this into collapse - usually.)

5. A math problem has never been solved. (After being solved, it takes surprisingly little time for new proofs to appear, compared to how long it took the original proof to appear.)

6. Being stuck in Earth's gravity well. Rarely, intense perturbations push things out into space.

Inspired by adamShimi's 'error correcting codes' here:

7. When writing out bibliographies by hand, errors in (copied) citations (proliferating). Or...ambiguity retention. Less ambiguity requires someone who has clarity, but if no one has clarity...

These are great! I particularly enjoy the social ones, like #1 and #3. They suggest a more general phenomenon, where people generally mimicking others can create all sorts of equilibria. It's interesting, because it's not like there's very strong incentives locking in those equilibria, just a kinda weak tendency, so shifting that sort of equilibrium could potentially be easier than the sort of incentive-locked equilibria which people talk about a lot.

2Pattern1moContents 1. Posts without comments 2. Endless threads 3. (Some) Posts with lots of responses 1. Posts without comments re: #1 A comment on something might spark discussion, but also People might read a post and have no comment. (Perhaps an upvote, but no comment.) A very long post could also have some of this effect by way of length, by taking time to digest, but it can also be something about the post (and the audience). (A work which is very good, and self contained, isn't necessarily a bad thing - it might end up shared a lot. Sometimes though, things just don't spark discussion.) 2. Endless threads A long time ago, on my shortform, I mused that whether a website was alive or dead could be pinpointed based on whether there was activity. More specifically, relative to a topic or subject - a post might not get responses, but someone might respond with another post. At one extreme end, a website could be alive, and have discussion without comments. On the other end, if there weren't anymore posts, but comments continued, it could still be alive. The main issues with that last possibility being: * Reddit inherently blocks this (threads get locked down after a month) * The format, UI, functionality etc., might have to be set up in a different way to facilitate that effectively. (If you're trying to look like Reddit, then while it might be possible, it's probably going to be a pain to navigate.) Other forum formats are a lot more friendly towards this. (For example, the responses~, or threads, are arranged in pages. (Thus say, navigating to the end consists of: scrolling to the bottom (click and drag the scrollbar, or use a key combination), and clicking the button with the last number.) * Abandoning infinite scroll, and things like it. * Might require some system for tagging people (or things one is responding to), or searching and filtering. * I could be wrong about this, but at least one orderly driver might be required.

The number of MHC class I loci. As the number of loci increases, the organism gains an ability to respond to a greater diversity of pathogens and avert evasion of an immune response. At the same time, with each new locus, any T cells that respond to self peptides bound to the new MHC class I molecule must be removed to maintain self tolerance. There is an optimal balance of MHC class I diversity and T cell count. In the case of humans, the optimal number of MHC class I loci appears to be 3.

1. The orientation of the shortest road between Chicago and New Orleans. As you progress along the road, you'll notice that it is mostly going south. Sometimes it jitters east or west but generally returns (and fairly quickly) to a southerly orientation. If you blasted a hole in some segment of road, making the shortest road something else, it would return to south-facing pretty quickly. (note: I was trying to think of one that doesn't involve time, at least not fundamentally.)

2. The position of my glass of water on the table. If I push down on it, it becomes ever so slightly lower even though it doesn't feel like it (the table is firm, but on atomic scales the atoms of my glass are getting slightly closer to it!). When I let go, it imperceptably springs back.

3. The number of unread emails in my inbox. It tends to 0 pretty quickly, until there is such a huge surge that I let a few go unread, and then there are X unread emails and that's the new equilibrium for a few months. (I suppose if you went and deleted one of them, then there would be X-1. But on at least one occasion I did this and then as a result slacked off a bit in my email-reading so that I ended up with X again.)

I love the first one. Explicitly trying to avoid time is a brilliant spin on the exercise.

  1. For any atom not being part of a living organism is a stable equilibrium.
  2. For a spinning top not to spin is a stable equilibrium.
  3. Hair length is a stable equilibrium.

Solid. I've seen a surprising number of people independently answer hair/beard length (including when trying this exercise in-person). Perhaps it is the most natural example of a stable equilibrium.

>!Certain math sequences that aren't very useful, like, to get the next number add the digits in this one. Should often get down to something stable.

The pre-Hadean earth as postulated: form oceans, suck up the CO2 into rock, cool down till the oceans freeze, stop sucking up CO2 and eventually volcanoes spit out enough it melts the oceans, etc.

Social popularity of certain things like, say, socialism, individualism/conformity, bowdlerism/pornography, anything where if you get too much of it it either blows up or at least people like it less.!<

Looks like I don't know how to do spoiler tags. Can't find it on site, alas.

2Ruby21dI just took a closer look at your comment, seems you tried the right syntax but it didn't mesh well with the list. I'll look into that tomorrow.