Perhaps the simplest way to picture this problem is to imagine a particle randomly moving around in a small box. The left and right sides represent different states, and your goal is to bring the particle to the right side. (5/n)
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If we had infinite time, the solution would be simple: apply the force infinitely slowly. But what if we want the result a bit faster? How should we apply the force to achieve this? (6/n)
It turns out that a general solution is very hard to find. Different systems react differently to the forces that are applied to them. There are some nice results for the case when the force is applied very slowly. (7/n)
But if you want results for the fast case, you really need to include in your analysis how the way you perturb the system changes the way your particle wiggles—and there are basically infinitely many ways for that. So how do we proceed? (8/n)
Very recently, there has been quite a breakthrough in addressing this problem: rather than looking at the optimal way to bring a particle from A to B in a fixed time (which is very hard), ... (9/n)
... people looked at the optimal way to bring a particle from A to B with a fixed average number of transitions between the states. This turns out to be a lot easier—but what does it mean? (10/n)
Remember that at this small scale, everything is constantly bouncing and wiggling. This allows you to think in terms of an activity rate: how often is the particle in my box moving from left to right (TICK!) and back again (TOCK!)? (11/n) https://www.youtube.com/watch?v=iP6XpLQM2Cs
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