 So if we start to model a small system and assuming that I have a few particles here, I might have a circle particle, I might have a square particle, and what should I use? Cross particle. Instead of having an actual surface here, what if I imagine that I have a neighbor system that I like a lot? The best possible neighbor system I could imagine would be one that is just as beautiful as we are. We have ourselves as a neighbor. So in that system I have the square, I have the circle, and I have the cross. That's not going to help, I just double the size of the system. Well, if I can accept having something that looks exactly like me, and that I can. I don't really need to simulate and integrate that. Whenever I need the position of that square, I know that it's going to be the position of that square plus the side of this box moved in that direction. So I can calculate the position of all these atoms from the positions of my central box here, meaning that I only need to simulate the central box, and then I can calculate the copies in the other systems. The trick here is of course that I'm going to do that for 26 periodic copies to the left, to the right, up and down, and in and out of the plane here. Sorry for drawing this a little bit sloppily. How large do these boxes have to be? Well, that depends a bit. I'm going to draw at least a few of them here. So we want them to be as small as possible, but not smaller. And the question is how small as possible is. So I only want to see one copy of each neighbor. And if I draw that here for this red square, I'm going to argue that that means that the cut of here, the interaction here, must be smaller than half the box, meaning that the box must be at least twice as large as the cut of radius I have. That will ensure that I only see one periodic copy of any molecules. I'm not going to interact both with that circle and that circle. That's important because it's not separate circles. They're really the same circles. They can't move closer because if that one moves down, that one moves down, etc. There is a name for that. It's called minimum image convention, but it's not important for this class. Then you will have to take the advanced simulation physics class. If we don't apply that, if you go to an even smaller system, what's going to happen is that you can end up with an effect that if this is now my simulation box, and there's a copy of me in that direction and a copy of me in that direction. If I push that copy of the professor standing over there, indecently enough, the second I do that, there's going to be somebody else pushing my back. That's rude. And if we translate that to atoms in the system, what's going to happen is that this atom, if at some point I have a fluctuation that I'm starting to move in that direction, what will instantly happen is that there's going to be somebody else pushing me from that direction too. And there's no friction. There's nothing stopping us. So for a system that's too small, you can end up with a phenomena called flying ice cube through space. So that this small system just ends up having greater and greater and greater velocity in one direction, and there is no friction stopping us. Trust me, that's bad. There are some ways around that, but somewhere there is a limit for how small we can make the system. But you can make the system almost that small. If you have a large protein, we typically add enough water around the protein. So this may be half a nanometer to one nanometer of water, if the cutoff is typically one nanometer. This means that I only need to simulate the central box. If a particle starts moving in this central box and gets to the boundary, well, the way I would like to describe this in physics is that it should just continue in the next box, right? And continue in the next box here just means showing up on the other side of my central box instead. So if something moves out here on the top side, it immediately reappears on the bottom side. My boundary conditions are periodic, and that is the name for this whole concept. Periodic boundary conditions, which we can abbreviate PBC. Very common concept in physics, really simple and it completely solves the problem. Did you see the other cool thing? There is no surface anymore. There is not a single water molecule exposed to vacuum or air here, so we can forget about the surface tension too. This is an awesome concept. If you don't believe my ugly drawing here, this is a slightly nicer illustration of it. It's exactly the same concept. Here you have a few more atoms and you can hit the pause button and entertain yourself a few seconds to double check that it is indeed correct.