William Blake and the ergodic hypothesis

Does your field have one very important fundamental concept that if it is not true your whole field is rendered meaningless? (And you won’t get a PhD and can open a bakery. Yay?) Computational biophysics has one, and it is called the ergodic hypothesis. Take it away, Wikipedia:

In physics and thermodynamics, the ergodic hypothesis^[1] says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time.

Uhm, crystal clear? Yeah me neither — I was scratching my head when I first encountered it. But rather than giving you a physics lecture, I thought as a civilised gentleman I will use (surprise!) poetry to explain it to you.

To see a World in a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour

— William Blake, Auguries of Innocence

I’m pretty sure William Blake was not thinking about the ergodic hypothesis when he wrote Auguries of Innocence (like 99% sure, we PhD students are never 100% certain, especially regarding when and whether we will graduate), but like your high school English literature teacher, I’m good at interpreting meanings that the authors never even intended to be. Ok, let’s not talk about high school literature (painful memories there, you know that I chose science in the end, right?).

Imagine a glass of water. We know from high school chemistry that energies of the water molecules are not identical, but rather they follow a certain distribution (Maxwell-Boltzmann distribution, if it is ideal gas). But the ergodic hypothesis says, that if you follow one particular water molecule long enough, it will eventually sample all the energies in the distribution. Kinda intuitive, don’t you think? But you see there are some problems there. How long is ‘long enough’? This is physics, so well, infinitely long, or eternity, though we can get good approximation in much less time (I have to graduate, you know.) This is dependent on the system, but usually a few hundreds of nanoseconds of simulation time is good for one solvated small protein molecule.

Why is it called hypothesis, not a principle, or something more solid, you ask? I think it is not a law or a theory because you can’t design an experiment to prove it. You see, it is like a postulate that so far works. Everyone in the field has started with this premise, and the things that we calculate are (usually) correct, as corroborated by experimental data, so we’d like to think it is true. Secondly, it does not always hold for any system there is, especially for exotic systems like a ferromagnet and those under extreme conditions, but for biomolecules in room temperature and pressure? It pretty much holds. Thirdly, maybe historically it was called that way and it stuck, as perhaps also the case for Avogadro’s hypothesis.

So now that you are clear I will try to rewrite the verse a bit:

To see a World in a water molecule

And a Heaven in a biomolecule

Hold good sampling convergence of Infinity in the palm of your hand

And a good estimate of equilibrated Eternity in a few hundreds of nanoseconds

(with apologies to William Blake)

You see now why I chose science and not literature?

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