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.

#### CategoryFirst-passage times

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.

**Basic strategies, timescales, and limitations**

**Basic strategies, timescales, and limitations**

Key biomolecular events – such as conformational changes, folding, and binding – that are challenging to study using straightforward simulation may be amenable to study using “path sampling” methods. But there are a few things you should think about before getting started on path sampling. *There are fairly generic features and limitations* that govern all the path sampling methods I’m aware of.

*Path sampling* refers to a large family of methods that, rather than having the goal of generating an ensemble of system configurations, attempt to generate an ensemble of dynamical *trajectories*. Here we are talking about trajectory ensembles that are precisely defined in statistical mechanics. As we have noted in another post, there are different kinds of trajectory ensembles – most importantly, the equilibrium ensemble, non-equilibrium steady states, and the initialized ensemble which will relax to steady state. Typically, one wants to generate trajectories exhibiting events of interest – e.g., binding, folding, conformational change.

The trajectory ensemble is everything you’ve always wanted, and more. Really, it is. Trajectory ensembles unlock fundamental ideas in statistical mechanics, including connections between equilibrium and non-equilibrium phenomena. Simple sketches of these objects immediately yield important equations without a lot of math. Give me the trajectory-ensemble pictures over fancy formalism any day. It’s harder to make a mistake with a picture than a complicated equation.

A trajectory, speaking roughly, is a time-ordered sequence of system configurations. Those configurations could be coordinates of atoms in a single molecule, the coordinates of many molecules, or whatever objects you like. We assume the sequence was generated by some real physical process, so typically we’re considering finite-temperature dynamics (which are intrinsically stochastic due to “unknowable” collisions with the thermal bath). The ‘time-ordered sequence’ of configurations really reflects continuous dynamics, so that the time-spacing between configurations is vanishingly small, but that won’t be important for this discussion.

The “Hill relation” is a key result for anyone interested in calculating rates from trajectories of any kind, whether molecular simulations or otherwise. I am not aware of any really clear explanation, including Hill’s original presentation. Hopefully this go-around will make sense.

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