 So based on Boltzmann, we're now going to change the way we talk about things, at least we're on the microscopic and the physical thing. We will no longer say whether things will happen or they won't happen, but we're primarily going to talk about probabilities. How likely is it for something to happen? Is it more likely to be at low energy and what is the likelihood of population of high energy? But doing things on an absolute scale is a bit complicated, right? I didn't even tell what the zero value here was. The Boltzmann distribution, and many of the other things we do, they're going to become really powerful first when we start to compare different states. So let's do that. Let's assume that I have a state A and a state B. What they are doesn't matter for now because we know that the Boltzmann distribution should be valid for both of them, right? So the probability of being a state A, let's say that P A, and here I'm going to use the constant. There is some constant, I just don't know what it is. That is equal to a constant, raised, multiplied by the exponential, raised to minus the energy in state A divided by kT. And similarly, we know that the probability in state B equals the same constant, multiplied by the exponential, raised to minus eB divided by kT. Now we don't know what those constants are, but a cool thing here now, what if I simply take the quotient of this? So what is the probability of being in state A relative to be in state B? That is, that will tell me if that quotient is larger than one, it's going to be more likely to be in A, and otherwise it's going to be more likely to be in B. If I now divide these expressions, those two C's, they will disappear. So that will be e raised to minus eA divided by kT divided by the exponential of minus eB divided by kT. Yeah, that doesn't exactly look simpler, but if you know your exponential and logarithm laws, if I have a quotient of two exponentials, that turns into the difference of the exponent, right? You all knew that by heart. So this I can write as e, and now I have to be careful with the minus sign. So the minus sign, I'm going to say minus eA minus minus. You know what? It's much easier to write this way. I will raise that the e minus eA minus eB divided by kT. That still has this eA's, but now I'll take the final step here. I can write that that's simply the exponential raised to minus delta eAB divided by kT. And the cool thing, if I now have an energy difference, it does not matter whether zero scale on my axis is right. I can put that 500 meters below the ground here. The difference is still the same difference. So the relative probability of being in two states, A and B, only matters on the absolute difference in energy between those two states. I was running a little bit ahead of myself here. And in particular, that means that if an energy difference between two states is very small, then this is going to be roughly zero, and then that quotient is roughly one. If there is a very large energy gap between them, what this tells is that the state here that has the lower energy is going to have almost all the population, while the one with higher energy is hardly going to have any population. So this always works. It doesn't matter if A or B is more likely. And we completely got rid of this constant in front of the exponential. This is pretty darn cool. And this will work in general. There is just one small problem here. This will definitely work for the simple column I showed. But what if I compare this? The column here is simple. I will have high density down here. I will have lower density far up. And any two levels I pick here, this will be valid for. But what if I want to compare that column with, say, that column? So now my approximation of unit volume as a function of height doesn't really hold anymore. I couldn't instantly guess which one of those is more likely, but maybe we should take two more obvious examples, triangles. Here I have hardly any volume at all down on the good level. Well, we have tons of volume up here where the energy is higher and the particle density is likely to be lower. Here on the other hand, I have plenty of volume down here where particles really want to be and very low volume up here. We don't know this yet and we don't know what it's going to look like. But I hope that your gut feeling also says that this is going to be better. Because in this case, we will have many more particles here because there is simply more room here. While here we hardly have any room, so there are going to be very few particles that can fit down here. The reason why we can't handle that is that my simple approximation assumed unit volume. Or rather, unit volume, a simpler way of saying this, at each height in that simple column, I kind of assumed that there is just one state. The state just depended on the height. So this only works if I'm comparing two specific states. But in many cases, there's going to be more than one place that I can put something in one state. At a very wide column, two particles can be on the same height. We're going to need to account for that difference. And it turns out that it's not going to be quite as hard as we think.