- Meteors. This one isn’t exactly right since the equilibrium point either happens or gets skipped over entirely but nonetheless...imagine a meteoroid flying through the solar system towards Earth. If it crosses into the atmosphere, it becomes a meteor. And then depending on the composition of the meteor and its size, it may burn up in the atmosphere or it may make it to the surface (at that point, it’s a meteorite).
In this case, you can think of the “zone of attraction” as the entire journey to the atmosphere. Once it hits the atmosphere, it quickly is decelerated by air resistance. And if the meteor burns up, then its velocity “stabilizes” at 0. So “burned up in the atmosphere” is sort of like an equilibrium point. If it doesn’t burn up (e.g. because it’s larger), then it plummets to the surface. In this case, the other side of the equilibrium point is not really repulsive. Rather, it’s just a continuation of the pre-equilibrium point. And then when it hits the Earth the entire system is changed.
- Nuclear Weapon Acquisition. You could argue the initial development of nuclear weapons triggered an “attractive force” for major world powers to stock up on warheads (e.g. an arms race). Eventually, the development and acquisition slowed down and even regressed (as in nuclear disarmament). But supplies were not totally eliminated. You could say an equilibrium point has been reached. The reasoning continues for the “repulsive force”: the equilibrium could be broken if any country were to deploy nuclear attacks on another. It would (probably?) lead to retaliatory strikes and the acquisition of warheads may accelerate drastically. (related: apparently the semistability of nuclear deterrence has a name! stability–instability paradox)
- My Steam library on a weekend night. On weekend nights when I have nothing going on, I am attracted to my steam library. I often arrive at an equilibrium where I will sift through my games but don’t actually play anything. Then, I either decide against playing games, at which point I usually back away from the equilibrium point for the night. Or some “external force” nudges me into playing, in which case +4 hours of my night are gone.
Unsurprisingly, friends-wanting-to-play-something-as-a-group is almost always enough to convince me to go past the point of no return. Similarly, I am also likely to break equilibrium if I am playing through a recently acquired game.
Again, I don’t think my first example is a true semistable equilibrium (neither the second example?) but maybe there’s something there. All in all, this was great. I just came across the Framing Practicum Sequences and love them! Thanks for the post AllAmericanBreakfast!
...now off to sift through some games :p
Meta-commentary: I found this frame bizarre. That's a good thing - I've never thought about anything quite like it before.
I feel like "semistable equilibrium" isn't quite the right way to frame the frame, but the examples were good enough to make up for that.
Minor editorial note:
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.
I think you probably meant to replace this bit with something else.
Finally: I had been thinking about offering a monetary reward for framing practicum posts which I find valuable. I hereby award this post $40. For AllAmericanBreakfast: message me a payment method to collect it. For everyone: this is not a commitment to do this for every framing post, but I will likely do this for framing posts which meet my highly subjective quality bar in the near future. I would have awarded $10 more if the post had a better-in-my-judgement name for the concept, along with a formalization which felt like a cleaner fit to the examples. I would have awarded $10-$20 less (but still nonzero) if the post had been worse in various other ways, but still generated interesting ideas when I did the exercise.
That's very kind of you! Thanks also for pointing out the edit - fixed! Incidentally, this example is taken directly from my differential equations textbook, which describes asymptotically stable, unstable, and semistable equilibrium points.
You may already be familiar with these concepts if you've studied this subject. If not, an asymptotically stable equilibrium point is a point at which velocity is zero, and that nearby velocities on both sides of the point move us closer to the equilibrium point (they are attractive). This tracks with your "stable equilibrium" frame. Unstable equilibrium points have zero velocity at the point, but a velocity that moves us away from the point on both sides (they are repulsive). Because of this, objects are rarely found at unstable equilibrium points, since any disturbance from that point will cause repulsion from it. An example might be a coin standing on its edge.
A semistable equilibrium point is a place at which velocity is zero, and where velocity tends to move us closer to the point on one side (attraction) and away from it on the other side (repulsion).
This is where the name comes from.
- Impact on a slab, like a bullet hitting armor or a rock hitting a windshield. If the projectile is long and dense enough to puncture, then it goes through and just keeps going - possibly even accelerating more, if e.g. it's falling. But if the projectile isn't past that critical point, it collides and either bounces off or blasts a crater in the surface or slows to a stop or whatever.
- When learning a subject, it often takes a lot of time to get to the point where it feels net-positive-value. Before that, each step is another point at which people "bounce off". But after that, people will just keep delving deeper and deeper.
- Similarly, a business might need to invest a lot of effort/money in order to become profitable. Each step before profitability has some chance of the founders/investors giving up. But once past the point of profitability, they'll likely keep investing more and more.
One thing I notice in the latter two examples is that we don't necessarily know where the equilibrium point is. We could maybe try to track "how close we are" by looking at e.g. a trendline of a company's profitability, although we don't necessarily know how "expensive" the next few steps will be.
I enjoy all these examples. They seem like examples of checkpoints, or milestones. Reconnecting with your examples of balls rolling on a ramp, we might view this as a ball with some initial velocity rolling uphill. If it can make it over the peak, it'll continue down the other side. Activation energy in a chemical reaction is a related frame.
One way to refine this frame further, and reconnect it with its origin in differential equations, is to consider a system in which the forces endogenous to the system reliably cause us to approach and reach the equilibrium point. In the absence of some external shock, objects will move toward this point, decelerate as they get close, and reliably stop when they arrive. This stands in contrast to a ball rolling uphill, which does not reliably stop when it reaches the peak of the hill, or move toward the peak if it is not pushed.
Nemoto suggested the example of a red light, and I think it's a good one. We slow to a stop as we approach it, and we'll stay there as long as the light is red. Once we're past the red light, though, we move away from it, as we do if some external force shoved us into the intersection when our light's red and traffic is approaching us on our left or right.
This seems like a case where it's particularly important to be clear on which forces are internal to the system, and which are external to it. Most real-world cases involve a complex mixture of forces. Semistable equilibria are delicate, since the smallest bump past them results in movement away from the point. In a system facing many disruptions, it may be hard to perceive these semistable equilibrium points, since they're disguised by these external disruptions.
Yeah, I'm familiar with the meaning in DEs, which is exactly why it seemed weird: the DEs version is the sort of thing which should basically-never happen, because it's extremely sensitive. Change the parameters even the slightest bit, and we either get two equilibria (one stable, one unstable) or no equilibrium.
Yeah, I think that’s right if we are considering a fast system and are being precise about the zero-velocity point. But the process of deceleration approaching zero at the limit could take a long time, and the forces that risk pushing us to the other side may be rare enough that we can often find objects at or near the equilibrium point.
Some properties that I notice about semistable equilibria:
- It is non-differentiable, so any semsistable equilibrium that occurs in reality is only approximate.
- If the zone of attraction and repulsion are the same state, random noise will inevitably cause the state to hop over to the repulsive side. So what a 'perfect' semistable equilibrium will look like is a system where the state tends towards some point, hangs around for a while, and then suddenly flies off to the next equilibrium. This makes me think of the Gömböc.
- A more approximate semsistable equilibrium that has an actual stable point in reality will be one that has a stable equilibrium at one point, and an unstable equilibrium soon after. I think an example of this is a neutron star. A neutron star is stable because gravity pulls the matter inward while the nuclear forces push outward. With more compression however, gravity overcomes these forces and a black hole forms, after which the entire star will collapse.
- War and Peace. Two rivals may sharply distinguish between threatening gestures and a clear attack. They can approach the point of no return many times, as a form of negotiation. But there is a threshold beyond which a fight or a war ensues: throwing the first punch, crossing the Rubicon, invading Poland.
- Flirtation and Breaking Up. Two people may approach the point of openly showing romantic interest in each other without crossing that threshold by giving a clear signal of their affection. Likewise, they may display signs of dissatisfaction with their relationship without explicitly showing that they are ready to break up with each other. Their attraction or animosity keep bringing them to the brink, but their hesitation about crossing the line holds them back as they approach it. There are gestures they can make which will cause an acceleration into romance or into social distance. But these moments "on the verge" are fleeting.
- Infection. Minor infections can sometimes be cleared up by the innate immune system without triggering an adaptive immune response. Beyond a certain point, immune signaling is sufficient to activate the beginnings of an adaptive response and trigger T cell proliferation.
1. Drilling a hole in glass. I was at a class learning glass fusing (just for fun) and we each had to drill a hole in float glass. The drill is a vertical bit, about 3mm diameter, and coated with an abrasive. 8 of the 9 of us in the class followed the instructions as to the angle at which to hold the glass, and to cool it with water frequently. We all cut neat little holes. The 9th person was in a hurry, and at the moment the drill broke through the surface of his piece of glass, it caught and violently span out of control, shattering.
2. This idea of a threshold point reminds me of what happens with exponential growth. In the early stages of the Covid outbreak, our Government were blithely aiming for "herd immunity" - after all, the graphs showed a gradual rise in cases, so everything must be okay, right? It took some serious educating to get them to see the nature of exponential growth, and that a disaster was waiting to happen. Now they seem to be taking the same view with the threat of inflation. Any exponential growth starts off slow and steady, like semistable equilibrium, but reaches the point where it is out of control if there is no intervention.
3. When I was reading this article, the image of a set of traffic lights came into my mind. On one side, you have a red light, with traffic approaching slowly and carefully. On the other is a steadily moving stream of traffic taking its turn to move ahead. In the middle somewhere is a point of equilibrium, where traffic waiting to turn right (I am in the UK; it would be left most other places) is paused in the middle of the road, having passed the red light, but being held up by the oncoming traffic. If one of these vehicles fails to obey the rules of the road, all kinds of chaos and mayhem could occur, with vehicles and other objects being flung and damaged unpredictably.
Interestingly, biologists have named the phases of the bacterial growth curve: lag phase, exponential phase, stationary phase, death phase. I suspect that having names for these phases helps people think more concretely. We could just say it's a variant of logistic growth, and give some sort of equation for it. But that wouldn't make it nearly as easy to talk or think about as naming the phases. Early in the pandemic, we could have been saying "we're in the lag phase of the pandemic, but the exponential phase is coming." Likewise, you're worried we're in the lag phase of inflation, and that the exponential phase is coming.
I wonder if we can just start talking like this. We know these trends are real, and perhaps the problem is that we talk about them as if they required an education in mathematics to understand. Instead, perhaps the bits of the graph just need some nice handy labels.
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Your graph also illustrates perfectly why I find this an example of semistable equilibrium as explained in this article. It even looks like a cliff face, although it is inverted. There is a point at which the lag phase changes and becomes the exponential phase. As long as the correct action is taken before this point, the exponential phase can be avoided; e.g. take the petri dish out of the incubator and put bleach in it. This would be equivalent to the chicken player stopping his car before the cliff edge.
Yep! On the flip side, biologists know that it’s important to passage cells before they enter the death phase, which means splitting them into multiple lower-concentration plates. Otherwise, the cells can be irretrievably damaged by overcrowding.
Thanks to John Wentworth for conceiving and executing the concept of a framing practicum, as well as much of the format and language of this post!
This is a framing practicum post. We’ll talk about what a semistable equilibrium is, how to recognize semistable equilibria 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 semistable equilibria 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 Semistable Equilibrium?
In Rebel Without a Cause, Jim Stark and Buzz Gunderson are racing their cars at top speed toward a cliff, with the gas pedal strapped down. The first to jump out of their car is chicken. Buzz's leather jacket gets caught on the door handle. He's unable to jump free, and plunges over the cliff to his death.
The cliff is a semistable equilibrium. The idea of the "chickie run" is that the cliff causes the racers to decelerate, so that their velocity approaches zero as they near the cliff's edge. That's how it works out for Jim. On the other side of the cliff, however, Buzz's velocity increases again as he hurtles toward the ground.
In general, a semistable equilibrium will approach an equilibrium point if it starts on one side, but will move away from the equilibrium point if it starts on the other side.
What To Look For
A semistable equilibrium needs a threshold that attracts and slows things down if approached from one side, but repels or launches things away if they're on the other side. If there's a point "point of no return," that may be suggestive of a semistable equilibrium. There's a zone in which disturbances lead to a return to rest, and a second zone just beyond leading to ongoing activity. It's possible to have multiple equilibria in the system. All that's required is that there is a point that attracts things from one zone, but repels things in another adjacent zone.
Whether or not it is common to find the system at the equilibrium point will heavily depend on the direction and relative magnitude of disturbing forces.
Useful Questions To Ask
Unlike with a stable equilibrium, the effect of nudges in a semistable equilibrium depend heavily on how close we are to the equilibrium point. If we're deep into the "zone of attraction" to the equilibrium point, nudges won't have much of an effect. But if we're near or at the equilibrium point, a small nudge could easily move us into the "zone of repulsion," leading to long-term instability in the system.
What happens if we change the equilibrium point? What are the disturbing forces in the system, and do they differ depending on where we are located relative to the equilibrium point? Can these disturbances "rescue us" from the zone of repulsion by bumping us back to the equilibrium point or into the zone of attraction? Does the zone of repulsion move us rapidly away from the equilibrium point, or is it a slower movement? How strong are the attractive and repulsive forces relative to any random disturbances in the system?
The Challenge
Come up with 3 examples of semistable 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?)
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:
Bonus Exercise: for each of your three examples from the challenge, what forces might allow you to predict, measure or control the approach or repulsion from the equilibrium point? Is there some intervention or disturbance that might push us in one direction or the other if we are near the equilibrium point?