 The simple way of introducing molecular simulations is just based on high school physics that you've all taken, but I'm going to write it down anyway. Newton's first law says that force equals mass times acceleration. So that if you know a force on a system, we can calculate its acceleration. But you know the force on a system. We've gone through that a couple of times in this class already. In general, force is the negative derivative of the potential, right? We can use the gradient for that in multivariate calculus. So if the potential line is mgh, the derivative that is going to be mg with a minus sign, which is exactly the force downwards. That's great. So if I just know all the coordinates, then I can calculate the force and then I can calculate the acceleration. But the acceleration in turn, that was the derivative of the velocity with respect to time. And that means that if I knew the previous velocity, I can calculate how the velocity is changing. And if I know that velocity, I know that velocity is the derivative of position as a function of time. So if I knew some starting positions here, and I know the velocities, I can calculate the new positions. And if I now have new positions, then I just go back here, plug those new positions in my potential and force. And then I keep repeating this for many time steps. And if I do this, and if I have accurate starting positions, I need to know the starting velocities too, I can just predict the exact motions, paths, trajectories individual atoms will take. There are a couple of problems with that though. First, this is going to involve some of the simplified potentials we used, we defined in lecture one, they're not perfect. Second, we do not know the starting positions exactly. Starting velocities, there is certainly no database where you can get that. So you're going to need to assign velocities maybe from the Maxwell-Boltzmann distribution, that's what I would normally do. But they're going to be random. And depending on the velocities you pick, you will have different paths here. That's going to start to sound a bit strange. The other problem is that even if you knew all these things perfectly when you start, the problem is that a computer is not perfect. The computer actually, we're using floating point numbers, 32 bits, 64 sometimes to treat this. And that means that there is a finite resolution to the calculation done by the computer. These systems are differential equations with many, many, many degrees of freedoms. And you can even show that they have so-called positive Lyapunov exponents, they are chaotic. What that means is that if I start with the system, if I start with exactly the same starting condition, and then I introduce an exceptionally small error, I'm taking just one bit in the computer. Initially, these trajectories are going to follow each other very closely. But eventually those differences will grow, and at some point I'm going to take a completely different path. So based on that, we would even say that there is no chance in the world that I can make absolute predictions of particles, even if I did know the starting conditions and the starting velocities and had a perfect potential, which I don't. So this is cheating a bit. I can't make absolute predictions in a computer about the positions of atoms. So we're going to define this in a slightly better way.