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.
I sympathize with #2, although for me the "titles become hard to read" issue is a lost cause - my equilibrium is way past that.
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:
And the bonus exercise:
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.
Most metrics of productivity/success are at a stable equilibrium in my life. For example:
Shifting equilibrium like these (being more productive, socializing more, getting in better shape, etc.) is obviously desirable. Let's explore that.
In cases like these, direct attempts to immediately change equilibria (like motivating myself to work harder in the moment, going to parties every night for a week, eating an unreasonable calorie deficit to "get fit quick," etc.) is like pushing the ball up the sides of the bowl. Why?
These equilibria are all determined by my own identity. The reason my [productivity/sociability/fitness] is [current level] is because I think of myself as someone who is at that current level of the skill. Making my [current level] jump in the short term does not change my identity and hence does not change my equilibrium.
The only way to "tip the bowl" is to change my identity, how I view myself. Probably the least-likely-to-fail way of doing this is in small increments. For example, instead of instantly trying to be supernaturally productive, first try cutting out YouTube/Twitter/Reddit. When it feels like this is an equilibrium, try reading a paper a week. Then a paper a night. Continue in small steps, focusing on internalizing the identity of being "someone who reads papers" so that the habit of actually reading the papers comes easily.
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.
!!! This got me so excited.
This is a knowledge gap I've always been missing about how the immune system works.
I love it and would love to get some sources I can read up on how immunity works, described in Mathy terms that I as a non biology person can understand.
Main exercise:
I recognize 1 and 3 are borderline dynamic equilibria but I think they changes on a slow enough timescale that they count.
Bonus exercise:
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.
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.
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.
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.
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:
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.
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.
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.
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.)
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.
(I may have pruned slightly too much)
1. Planets orbiting the sun: Little disturbances will still result in an elliptic orbit, you need a big disturbance to get the earth fall into the sun or have enough kinetic energy to escape the solar system on a hyperbolic (or in the edge case parabolic) curve.
2. People don't walk naked through streets, because when you try it, you get punished for it. (and other psychology reasons)
3. Your neurons firing in stable similar patterns that make you able to walk upright instead of falling over. Even if you stumble you can often catch yourself because your neurons fire in patterns that are optimized for you standing/walking.
Bonus exercise:
(The factors I can ignore are basically named above.)
Changing the equilibrium:
1. Another star flying through our solar system.
2. Ok "not naked" arguably isn't a that special case out of the possibility space, so it doesn't buy you that much to describe it as equilibrium. There are still many possibilities of how you can be not naked. Still, it is quite non-trivial to get into the "walk naked on the streets" state and you'd probably need an intelligent actor who really wants to achieve that.
3. Being pushed hard. Getting signals from other brain areas that you should sit or lie down.
The distance between a car and its garage. The car tends to be parked at the same place every night, so it returns to 0, despite some perturbations that may happen. However, the car might eventually be moved to a different long-term garage.
Blood glucose levels. I think this applies to a lot of biological systems, actually. But blood glucose level will "spring back" from local perturbations.
Price of a commodity. There can be surges in demand and supply, but generally the price should stay in the same place.
Regarding (3) suppose we want to change the equilibrium (make the network classify the image correctly). Perhaps you could apply the same methods used to create adversarial examples to find the smallest perturbation that makes the network classify the image correctly, then maybe you can do interpretability stuff on the perturbation? or look for systemic errors by finding patterns in the perturbations? To make this work well you'd need some vision meaningful definition of "smallest perturbation" I think
I've been thinking about entropy a lot these days, not just in the usual physical systems with atoms and such sense, but in the sense of "relating log probabilities to description length and coming up with a way of generating average-case short descriptions, then measuring the length of the description for the system and calling it entropy". So I might just run wild with it.
Lies and manipulation in large organizations. This tends to an [immoral maze] / [high simulacrum level] equilibrium where people don't talk about object level things and mostly talk about social (un)realities. This is related to entropy and shortest length descriptions because there are more ways to talk about social realities than there are ways to talk about object level truths.
Physical entropy. This equilibrates at a maximum that's related to how big / complex the system is (how long of a string it would take to describe the least likely state in the system when you use a system of descriptions that tends to produce shortest length descriptions). Similar to the previous case, this has something to do with how there are many, many states with low likelyhoods and long descriptions.
Life (as in, cells that are lit ("alive") and move in complicated and interesting ways in Conway's game of life. They tend to get locked into repeating patterns or die out. I don't have a good intuitive explanation for this, just that there are a lot of ways things could die, and not many ways things could come alive.
I would love it if somebody could critique my examples and help me get a deeper understanding of entropies and equilibria. I have a vague intuition about how, in order to count states and assign probabilities, you really really need to look at how state transitions work, and how entropy is somewhat related to some sort of "phase space volume" that isn't necessarily conserved depending on how you're looking at a system. I feel like there's probably a lesswrong post I haven't seen somewhere that would fill in my gap here.
If there isn't, I would love to get some encouragement and write one
Examples:
the number of popular X in a human system Y:
Orbits just came to me, not sure if that counts a novel but I had never thought of them before as a stable equilibrium. They should stay the same unless perturbed by an outside force... but now that I think about it, pushes on an orbit are a permanent change. So I think that changes my answer to a non-stable equilibrium.
It feels to obvious, but fungible, replicable, commodities equilibrate sales price = MR.
Political environments should be stable as well until someone changes the system which created them. I'm labelling this as qasi_stable, they find a local minimum based on the rule set, but external forces can eventually break the system (see: all historical empires)
>!
The Pruned
Epistemically Uncertain: Gender norms. I thought about this one for a while, and it was interesting but I just don't see too many equilibrium forming around "stable" femininity or masculinity throughout history. oh well. I also removed the Lipostat.
Boring Repetitions: Temperature of human body, other biologically important homeostasis levels, or psychologically important set - points. !<
I can't seem to put spoiler tags on this?..
>!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.!<
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…).
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.
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:
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:
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!
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: 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.