This is a framing practicum post. We’ll talk about what timescale separation is, how to recognize timescale separation 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: pick 3 examples of equilibria from the previous three practica, and for each of them, give one use-case on a fast enough (or slow enough) timescale that we can treat the system as constant (or in equilibrium). 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 minutes at most.

What’s Timescale Separation?

If I put a piece of iron in the ocean, its equilibrium state is rusted through. However, it takes a long time to reach that equilibrium. If I swim past and look at the piece of iron, I probably won’t even see it gradually becoming rustier.

At the timescale of a person looking at the piece of iron on the way by, its state is roughly constant, even though it’s out-of-equilibrium. The changes are so slow that we can ignore them.

On the other hand, consider pushing a wheelbarrow. The force isn’t transmitted to the wheelbarrow instantaneously - for a brief fraction of a second after I push on the handle, the handle actually compresses a bit, and the compressed handle presses on the wheelbarrow frame, which compresses the frame a bit, which then presses on the load… the compression propagates as a wave, transmitting the force through the whole wheelbarrow. And the wave also bounces back, exerting pressure from the load back on my hands. But from my point of view, this whole process happens extremely quickly. Within a fraction of a second, the waves have settled down to an equilibrium force (and matching acceleration) between my hands and the wheelbarrow.

At the timescale of a person pushing the wheelbarrow, the forces are always roughly in equilibrium. The force-propagation process is so fast that we can ignore it.

This is timescale separation:

• If a system equilibrates on a timescale much slower than whatever-we’re-interest-in, then we can approximate it as being in a constant non-equilibrium state.
• If a system equilibrates on a timescale much faster than whatever-we’re-interested-in, then we can approximate it as always being in equilibrium (even if the equilibrium changes slowly over time).

Note that a system may involve multiple processes which equilibrate on different timescales. For instance, in the wheelbarrow example, there’s a very fast equilibrium of forces between my hands and the wheelbarrow, but also a slower equilibrium in which I set a steady walking pace.

One common pattern to watch for: often a system doesn’t return exactly to equilibrium, but exponentially decays toward equilibrium. (In general, this happens whenever the rate-at-which the system moves toward equilibrium is proportional to its “distance” from the equilibrium state.) In this case, the half-life is a good “equilibration timescale” for purposes of Fermi estimates and timescale separation.

What To Look For

Timescale separation should come to mind whenever we have a stable equilibrium. For any equilibrium it’s worth asking:

• Fermi estimate: how fast does the system equilibrate? If it decays to equilibrium exponentially, what’s its half-life? (Note that there may be multiple processes in the same system which equilibrate on different timescales.)
• What things am I interested in which happen much faster than the equilibrium?
• What things am I interested in which happen much slower than the equilibrium?

The Challenge

(Rules adapted from the Babble Challenges)

Pick 3 examples of equilibria from the previous three practica, and for each of them, give one use-case on a fast enough (or slow enough) timescale that we can treat the system as constant (or in equilibrium).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:

• First, estimate how long it takes the system to equilibrate if it’s “poked” somehow.
• Next, think about ways you might interact with the system much faster/slower than that.

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 post is meant to practice the “action” step: once we recognize an equilibrium, what questions should we ask or what approximations should we test?

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