Count me among the weak of science.
Here I am again, feeling defensive, irate at reviewer critiques of our recent sub-Nobel prize work. Only in this case, the reviews are in my mind, yet to arrive. In fact, we haven’t even drafted the paper yet! But I can foresee what will happen. After all, if I’m honest, our contribution is clearly incremental.
Is it a failure? Were my expectations way off, again?
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Many scientists say they dislike writing, but I want to persuade you not to be one of them. Writing not only is the way to convince folks how important your work is, but it can be a key part of doing good science in the first place. Your work must be explainable in a concise and logical way … or else it may not be logical!
And writing can be fun, once you realize it’s another more-or-less scientific puzzle to solve: how to explain what you’re doing to a target audience (or audiences). What’s the one-sentence version of your work? The three-sentence version? How would these change for more general or more technical audiences?
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Did you know that you can significantly change women’s performance on a challenging math test simply by providing a different explanation of the purpose of the test up front? And the same for Black students? Did you know there are ways to give advice and feedback that are demonstrated to improve performance in groups that suffer from (unconscious but inevitable) internalization of stereotypes? I learned these facts just a few weeks ago and find them astonishing. How can these crazy but simple things be true, and more importantly, why don’t most of us – professional scientists – know about them?
The key lesson of all these exercises is that you can push yourself to be better and more confident in theory by tackling simple, paradigmatic problems in an incremental way. You must put pencil to paper! You must do it regularly. But once you do, the benefits come quickly. Each mini-realization builds into knowledge. Each solved simple problem builds your intuition for understanding complex systems.
Give yourself a pat on the back if you’ve come this far. You have used simple exact solutions to differential equations to grasp the essentials of non-equilibrium processes. But there’s the physical process on the one hand, and the mathematical description on the other. We’ve used continuous-time math thus far. We now move to discrete time and get a taste for “Markov state models,” which implicitly employ time discretization in the field of biomolecular simulation.
I think of knowledge like a house: it’s assembled from bricks that separately don’t do much. On its own, each brick is more prone to weathering. Likewise, each of our calculation bricks is easy to forget. We must learn to put these in context. We must always seek the connections to build a stronger house, which automatically preserves our individual bricks.
If the two-state system is the hydrogen atom, the three-state system is the hydrogen molecule. We have plenty more to learn about the three-state system. Mastering this material will really boost your confidence with non-equilibrium systems. Of course, we already studied the two-state system when it was out of equilibrium: remember the relaxation time 
? But that was relaxation to equilibrium. Relaxation to a non-equilibrium steady state (NESS) is more interesting.
Three states – now that is exciting! And I’m not kidding. You are poised to understand critical timescales in non-equilibrium statistical mechanics.
Remember our goal is to crawl before we walk. We want to absolutely master the basics, so that complicated systems are less incomprehensible. While it is essential to understand the two-state system, it’s not enough. Using three states will deepen our appreciation of relaxation phenomena and timescales. And we will take the first step toward understanding non-equilibrium steady states.
To become stronger in theory, we do math. But we need to understand mathematical results in physical terms. The math and the conceptual picture must reinforce one another in our minds, or we’ll forget both. We also have to understand the assumptions underlying solvable models.
So it’s time for the solution to the previous problem … and new questions to understand that solution.
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