 So we know if we can write down all the energies that a system has, we can use those to calculate the partition function, which is the sum of all the Boltzmann factors for the system. But that calculation is made a lot easier, it turns out, in a case that's pretty common, which is sometimes we have one large system which is composed of multiple smaller parts or multiple independent subsystems. So let's suppose we have a system whose total energy can be written as the sum of the energy of some part A and some part B. And there's a lot of different occasions when that might be the case. We might have a box with multiple molecules in it. Let's say we have a box with two molecules in it. A is molecule one, B is molecule two. So the energies of the two molecules sum together are the energy of the whole system. Or sometimes we might prefer to break the energy up into kinetic energy and potential energy. Or in different circumstances we might prefer to break the energy up into different flavors of energy, like the translational energy, the energy that something has when it's moving, and the energy it has when it's rotating, or the energy it has when it's vibrating. So the different flavors of the energy, there's a lot of different ways to break up the energy. But if we can break up the energy into two parts, and an important caveat, if those energies are independent, meaning if the energy of one part A of the system doesn't have any effect on part B, then we'll see there's a shortcut we can use to write the partition function. So what would the partition function normally be? If we didn't think about breaking it up, that would be just the sum of all the Boltzmann factors, e to the minus energy over kT. And let's say that's a sum of, instead of using i this time, I'm going to use the subscript alpha, but again, this is the sum over all the states of the system. Energies of those states over kT, up in the exponential with a negative sign, that gives me the partition function. But I've just said that if the total energy of the system can be broken down into two parts, then of course we can break down this total energy of the whole system into the energy of part A and the energy of part B. The sum is the sum over all the states that the, in this original description, the sum is over all the states that the entire system can have. So part A, maybe it's a molecule, can have energy levels one, two, three, four, five. Part B, same type of molecule, different type of molecule, can have its own different energy levels. So the sum of all the possible states the system can occupy needs to include all the states i that part A can have, as well as all the states that I'll label j, that subsystem B can have. So maybe if they're different molecules, the energy levels that molecule A can have are different than the energy levels of molecule B. So I need to account separately for all the ways that A can have some energy and all the ways that B can have some energy. So this is our new sum over all the possible states the system can be in. System A is in state i, system B is in state j. So now I've just written the total energy as the sum of these two energies. But of course, e to the sum can be decomposed as a product of two exponentials, so I could write that as e to the minus e a over k t times e to the minus e b over k t. Remembering that I'm summing over all the states i that system A can be in, all the states j that system B can be in. But there's no reason to keep both of these exponential terms inside of both of these sums. The sum over j only applies to system B, and the sum over i only applies to system A. So I can pull this first Boltzmann factor out of the second sum because it doesn't depend on j. And when I rewrite the expression now, I've just essentially exchanged the position of the Boltzmann factors through one of the sums. But notice that what I can do with this expression is that it's, instead of written here as a double sum over these products, now I've written it in a way that can be interpreted as the product of two sums. If I just stick parentheses here, we see that this is the product of a sum of Boltzmann factors for system A, and a different sum of our Boltzmann factors for system B. So we know what a sum of Boltzmann factors is called. That's the same thing as a partition function. So this first term in parentheses, the sum of the Boltzmann factors e to the minus i over k t for all the states i the system A can occupy. That's just the partition function for subsystem A. If I were to have from the very beginning calculated what is the partition function for this part of the system, that's what I would have gotten. Likewise, the partition function for system B that can occupy any of these j states is the sum of its own Boltzmann factors. So I can write that second part as the partition function for system B. So what we've seen is the partition function for the whole system. If the energies are independent and if the energy can be decomposed into the sum of two different energies then the partition function for the whole system can be written as the product of the partition functions for the two smaller systems. So that's a pretty important conclusion. We'll see soon that it's much easier to calculate the partition function for a small portion of a system than for the entire system as a whole. And this lets us calculate the big system partition function from the product of these smaller partition functions. A few comments. It looks like I've been inconsistent here with writing q as a capital Q or q as a lowercase q. And that's going to be typical. When we use q to represent a partition function, it doesn't matter much whether we use a capital Q or a lowercase q to describe the partition function. Usually what I'll do is I'll use the capitalization of that to give some hint for whether the system is a large system that contains multiple smaller systems or a small system. When we calculate the partition function for a single molecule, for example, I usually use a small q. When we calculate the partition function for a large system like a box of molecules, then I'll typically use a large q. But it doesn't matter. That's just a semantic hint. It doesn't change the fact that both of those are being used to describe the same quantity. A partition function is still just a sum over Boltzmann factors. So the other note that I want to point out is we can use the same trick if we have more than just two subsystems. If we have two subsystems A and B, if their energies can be summed, then their partition functions get multiplied. If the total energy is energy A plus energy B plus energy C for something that can be broken up into three different subsystems, three molecules in a box, then it probably won't surprise you to see that the partition function will be the product of the smaller partition functions for subsystem A times that for subsystem B and that for subsystem C. And we could repeat the algebra, but the easy way to understand that that must be true is, as a first step, I could decompose the system into A and all the rest of it. And I'd get a partition function for A times whatever the partition function would be for the rest of it. And then the partition function for the rest of it, I can then subdivide into these two parts. So in general, no matter how many pieces I can subdivide the energy down into, the partition function is going to be the product of all those smaller partition functions. So this is a very useful trick when we go to calculate the partition function for large systems, being able to calculate it in terms of smaller pieces. The most common example while we're do that will be if a box contains multiple molecules, very often a very large number of molecules. And so there's an additional simplification we can make if all these molecules are identical to each other, and that's what we'll consider next.