 So it gets a bit tedious to keep to have to write those proportionality factors all the time. So what is that if I could say that the probability of being in state A, well we know that that should be proportional but say let's say equal to, let's say that is the e raised to minus eA divided by kT. Couldn't I just normalize this by summing over all states and e raised to minus eI divided by kT. Yes, if I know all the states I can do that. And this is so common that in physics we call the entire denominator here for Z, the partition function. That might seem like a stupid or obvious result, it's just a sum. But you know what, it's not so much the sum that's the key thing that, but the process of performing the sum requires you to have information about the energy of every single state. And that will of course mean that we know the relative weight of each state. So if we know the partition function that means that we know every single microstate in the entire system. And then we know everything about the system. Because let's say that I would like to calculate something specific. I might want to get the energy, the energy of the total energy of the system. Well, that just corresponds to the sum of the energy in state I, multiplied by the weight in state I. I can call this the weight of state A divided by the partition function to normalize it. This goes for any property that I can calculate in each state. I know what the energy of state I is, right? So if you want to average anything, say over molecules or something, as long as you can enumerate this over states, you're done. Statistical mechanics will tell you everything. This is great in a world where you're playing with toy models and you have four states. Unfortunately, we tend to have more than four states and things we're working with such as proteins, like a quadrillion states. This is not possible to enumerate even for a computer. So on the one hand, it won't work. But if we look at this, there will be some weights here with very high energy. And if the energy here is very high, the weight is going to be very low. We can probably ignore those. And the trick here is that if I know the partition function exactly, then I can calculate all these properties exactly. But if I know the partition function approximately, I can calculate things approximately. And in many cases, computer simulations will work quite fine because they're much better than I am at sampling things. And if the computer simulation can at least sample the 90% most relevant parts of space, then I can get something that has a pretty good approximation. And you're going to be doing this later on in the class, although you won't get anywhere close to completely sampling a system. But any time we're doing computer simulations in physics, where you, at least statistical mechanics, it's not necessarily a matter of predicting a trajectory, but you're really sampling states from the partition function. One thing that we would like to calculate, though, is entropy. And the problem is that what is the entropy in a particular state? We don't know, because that has to do with the probabilities of organizing things in that state. And if I put the molecule in this state, how many molecules can go in the other states? And to derive that, we're going to need to look at one second derivation here where we involve a bit of equations. But it's not going to be super difficult.