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.
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Relaxation and all that (Exercise 4)
Three states – now that is exciting! And I’m not kidding. You are poised to understand critical timescales in non-equilibrium statistical mechanics.
From two states to three (Exercise 3)
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.
Squeezing physics from math (Exercise 2)
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.
Becoming theory-strong, at least stronger (Exercise 1)
With this post I want to begin a series of exercises designed to grow your strength in theory pertinent to statistical biophysics – i.e., in math, physics, theoretical chemistry. The goal is to help you find a sweet spot where you push yourself a little bit, and regularly so you can continue to improve. Along the way, you’ll (re)learn critical statistical physics, which will help you understand, implement, and assess methods and findings more effectively. Of course, you’re in!
Do we have a gender problem?
Ten days after I made this post, the New York Times published a related piece suggesting that one gender was more likely to up-sell their work with self-congratulatory description. (Guess which one?) This made me think more about the gender issue in our own field. Have something to say on this issue? Write me: zuckermd@ohsu.edu. I hope to do a post on this in the future, ideally relating the experiences of several individuals, anonymously if they wish. I will be at the Biophysics meeting in San Diego Monday and Tuesday if anyone wants to talk about it.
Let’s fix MD – You can help
It’s my view that we must become statistical biophysicists. Why statistical? Because microscopic behaviors must be repeated zillions of times to create macroscopic effects. Can you help me shift the thinking in our community? See below for a collaboration opportunity.
Should we believe your abstract?
How many times has our community solved the sampling problem? I think it’s a fair question. You know I’m talking about claims rather than actual solutions. And many if not most of those claims are made in the abstracts of papers, even when the data paints a more limited story. I think our abstracts are the problem.
Learn your lesson: Every method has its limits
It was at breakfast during a recent conference that Prof. X leaned toward me and quietly said, “We tried using weighted ensemble but it didn’t work.” I got the sense he was trying not to broadcast this to other conference attendees, as a courtesy.
A “proof” of the discretized Hill Relation
This is yet another one of those things where, after reading this, you’re supposed to say, “Oh, that’s obvious.” And I admit it is kind of obvious … after you think about it for a few minutes! So spend those few minutes now to learn one more cool thing about non-equilibrium trajectory physics.
In non-equilibrium calculations of transition processes, we often wish to estimate a rate constant, which can be quantified as the inverse of the mean first-passage time (MFPT). That is, one way to define a rate constant is just reciprocal of the average time it takes for a transition. The Hill relation tells us that probability flow per second into a target state of interest (state “B”, defined by us) is exactly the inverse MFPT … so long as we measure that flow in the A-to-B steady state based on initializing trajectories outside state B according to some distribution (state “A”, defined by us) and we remove trajectories reaching state B and re-initialize them in A according to our chosen distribution.