 But even with a fast computer, you will never be able to exhaustively sample everything. And we've talked about that concept already. Remember the energy landscapes. In this lecture you will occasionally hear me talk about phase space, and it's really the same thing, but the energy landscape typically only refers to the positions of all the atoms in the system, while the phase space also takes their velocities or their momenta into account. In practice for proteins, we rarely care about the second part, so you can use phase space and energy landscape almost interchangeably here. But the point is that if I only run a small simulation, I might stay around this small blue part here, and that's frequently true. Let's minimize that and show you such a simulation. Small tripeptide, I think it is for maybe alanines that are rotating. I think I'm showing maybe 10 or 20 picoseconds here. You see that it's kind of oscillating, but not really sampling significantly different conformations. At some point here, I think you will have one or two of the back bone torsions change, so maybe we're moving over to that second small blue part, but I'm certainly not exhaustively sampling all possible states here. And that's to be expected. We will definitely not sample the states where two atoms are colliding and overlapping, because that energy would be astronomically bad. So on the one hand, that means that even if I were to continue the simulation 100 times longer, I still wouldn't sample the red parts of the energy landscape. And if I'm not sampling that, I'm not really reaching the partition function, right? And that was what all this was about. You remember those parts? That the weight, the probability of finding the system in state i, that was e raised to minus the energy of that state divided by the partition function capital Z. That was the sum over all such energies. And the power of that is if I know that I can calculate the average value of a property A, I'll get back to what those brackets mean, as the sum over all states, the weight in each state, multiplied by the value the property had in that state. But if I can't do that sum, I can't apply that. Well, maybe I can. Not exactly, but what if I apply that approximately? The problem here is that I might only be sampling 1% of the states. So what if I throw away 99% of all possible states here? That would be horrible. Well, yes, if I had no other information, that would be horrible, but I'm not throwing away any states. So the key idea here is that what if the 1% of the states I have are all the blue states here? Those are going to be the states. It's just 1% of the states if I'm counting, but they might correspond to 99.9% of the weights. Those are all the states that are going to have large weights and high population. So sure, I definitely ignore 99% of the states if I count, but those states are going to be the red ones. So although in absolute numbers it might be a very large states, the total weight of those states might just correspond to 0.01% here. And if I'm ignoring a very small part of the weights, even if it's lots many high states, that average is going to be pretty accurate anyway. And that's how it works in practice. If long as I sample, my simulation samples the relevant parts of space around the regions I'm interested in, if I do that sampling well, I'm going to get reasonably good averages here. These brackets typically indicate an average over phase space or the ensemble, and I haven't really defined what an ensemble is, and I'm going to need to do that because that influences some things. For instance, the molecule here is moving. Why is the molecule moving? Somebody gave this molecule a velocity. Yours truly gave it a velocity. I decided that this molecule should have 300 Kelvin velocity, but that was my completely arbitrary choice, and where do we define what these choices should be? I'll show you.