Let's do this
Bonus exercise
I really like the chess example. Anything continuous that gets discretized is similar. Like the color of a pixel in a photo, or whether you have crossed the finish line in a race.
Chess example is awesome, I had never of that before. Like, if I set down the piece overlapping the edge of a square, I will want to push it unambiguously into the square.
Good economic examples in 2 & 3. I find 1 particularly novel - it works surprisingly well as a bistable equilibrium, with the "in recycling bin" equilibrium more stable. If a container is near that state - e.g. it's mostly empty and sitting around in some random spot - then usually someone will empty it and throw it in the bin.
I find 2 particularly interesting, because it matches my experience, but I have no idea what mechanism drives the system into discrete-ish states. Now I think about it, clouds seem related: we often see a "partly cloudy" sky with lots of discrete clouds scattered around and empty space between them, rather than a uniform less-concentrated cloudiness throughout the sky. That suggests bistability in cloud formation. What's up with that?
I've thought before about onboarding new users to an app in a similar way to #2.
#3 ties nicely to the "mental mountains" model - in particular, we might guess that psychedelics lower the "hump" between two stable states, so it's easier to switch tastes/habits.
1. When my alarm rings in the morning, I can collapse into one of two possible equilibria: Either I stand up, or I hit snooze and continue to lie in bed for another couple of minutes.
2. In Schrödinger's cat example, either the atom will decay or not, so the cat will either be dead or not, and it doesn't make sense to describe it to be in a superposition. (Schrödinger's cat is terribly often totally misinterpreted by the media and others.) (Admittedly the cat already is in one equilibrium and needs a kick to get into the more stable equilibrium of being dead, so this example is bad.)
3. For a particular type of tech startups, those startups will either become relatively big successes or fail eventually (at least most often).
Bonus:
Exercise
Bonus exercise
1) As a young man I spent some time jumping back and forth between being single and being in a relationship. Both states felt stable, like two distinct equilibria.
2) Also as a young man I had friends who pursued higher education and friends who didn't; then there were those who alternated between the two states, once or twice or more. Perhaps not the best example because college is not an ending state but a beginning stage.
3) I'm reminded of the first two lines in "All My Love" by Led Zeppelin: "should I drop out of my farmer life / to chase a feather in the wind". Both states, farming and being a merry gypsy traveler are states of equilibrium, in their own way....
All of my examples are similar, I know. The first thing that struck me upon reading the post was the human aspect of this concept, how a person's life can have more than one equilibrium, depending on your starting position and trajectory. Bonus below:
1) She breaks up with me! Or vice versa. Or I'm walking down Grosvenor Square and a girl catches my eye, with bells on her fingers and rings on her shoes. That may set my ball rolling toward the hypothetical equilibrium of her arms. I might as well try.
2) COVID-19 pushes classes online and a struggling student decides he should take the leap into the professional world instead of the academic.
3) I think Robert Plant was singing about something similar to my answer for number 1).
I wonder if we can think of a physical metaphor for an inversion of this, when pushing harder on one pole lowers the transition cost such that a sudden flip becomes more likely.
This is a framing practicum post. We’ll talk about what bistability is, how to recognize bistability 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 bistability 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 Bistability?
The classic picture of bistability is a marble in a double bowl:
The marble has one stable equilibrium on the left, and another on the right. Although the marble has a whole continuum of possible positions, when left to its own devices for a while it will settle down to one of just two positions.
My own head-canonical examples of bistability come from digital electronics. One is the signal buffer: it turns a sorta-low voltage (like 1 V) and into an unambiguously low voltage (like 0.01 V), or a sorta-high voltage (like 4 V) into an unambiguously high voltage (like 4.99 V). One stable equilibrium is at 5 V, the other is at 0 V, and all other voltages get pushed toward one of those two. This is crucial to building large digital circuits: without buffering, 5 V would decay to 4 V then 3 V as we pass through one gate after another, and eventually we wouldn’t be able to tell whether a voltage is supposed to be high or low.
Another electronic example is the latch, one of the standard low-level memory elements in digital circuits. You can think of a latch sort of like the marble-in-a-double-bowl, but with two extra features:
So, to “write” a bit into the memory element, we push the system into the desired “bowl” (i.e. basin of attraction). It then stays in that state indefinitely, and we can read out the stored bit as many times as we like until it is “overwritten” (i.e. the state is set again).
What To Look For
In general, bistability (and multiple stability) should come to mind whenever an analogue system (i.e. a system with continuous state variables) has discrete behavior. In particular, it’s usually necessary for lossless transmission/storage of discrete information - a system with a single stable equilibrium has no long-term memory, since it always returns to the same state.
Useful Questions To Ask
In the double bowl picture earlier in the post, there’s a hump between the two stable equilibria. The higher the hump, the harder it is to push the marble from one equilibrium to the other. In chemistry, we call that height the “activation energy” - the energy which must be provided to move from one stable state to another. If we want to switch the marble’s state (e.g. to write a bit into memory), we need to provide that activation energy, and a “higher hump” makes it harder/more expensive to set the bit. On the other hand, a higher activation energy makes it less likely that random noise will accidentally push the marble from one side to the other, so a higher-hump memory element can store a bit for longer.
In general, other than the usual equilibrium questions, in a bistable system we usually want to know what’s required to change from one stable equilibrium to another. A few ways this can apply:
The Challenge
Come up with 3 examples of bistability 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 may 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 kinds of “kicks” could cause the system to switch state? What controls how strong the kick needs to be? Suppose you want to switch the state, or want to know what caused a state switch; what kicks could you rule out on the basis that they’re too small? Can you do a Fermi estimate for how often a kick large enough to force a state switch happens due to random noise?
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 bistability.
Hopefully, this will make it easier to notice when a bistability frame can be applied to a new problem you don’t understand in the wild, and to actually use it.