 So let's have a closer look at the Boltzmann distribution and, in fact, even derive it for a special case. The reason for that is that I actually find it easier in the general case because there are fewer things I have to define, but the first time it can sound very abstract. And my experience says to me that most of you find it easier to understand the special case. Hopefully, this wets your appetite so you see the power of the general case. Just as Finkelstein, I'm going to use the case of I have a column of gas, whole column, and then I want to study the distribution of these gas molecules as a function of the height here, h. Now, in general, there are two components acting here. We have a pressure acting up, and we have the gravity, which is the component forcing all the atoms down. The pressure forces to spread out. Otherwise, everything could be down here. If there was no gravity, we would have equal density everywhere. But with the gravity, we need more molecules down here when the potential energy is better. If I look at the particular thin slice here, under equilibrium, the difference if I move up throughout this slice, well, I can calculate that at a particular value, roughly from the derivative multiplied by the thickness of the slice. Or, alternatively, if I move up this slice, that means that what is the extra pressure? Well, pressure is forced per area. So the extra pressure or the difference in pressure I get is going to be the force for all the atoms or molecules in this thin slice and how much they contribute to the pressure downwards, and that I can get from the change in the potential energy. So I just need to derive those two expressions, then put them equal, and then try to solve the equation and see what that should be. The pressure is probably the easiest one for you. Pressure, gillous access, pressure multiplied by volume is nKt, where n is the number of particles. I could work with this, but then I need to start worrying about what the volume is, what is the absolute number of particles I'm interested in principles. And because I have the same area here all the time, I'm not really changing any volumes, so I can just choose to introduce lower case n, which is the number of particles per volume as a function of the height, and that would just be nKt. So lower case n is upper case n divided by volume. I haven't lost any general. In this equation, k and t are constants. p and n are functions of the height. I can take the derivative. So dp dh equals dn dh kt. I'm not sure if you've taken the last analytic mathematics, but as long as I'm working with functions of a single variable, I'm actually allowed to separate this dn and dh, or dp and dh here, and use them just as if there were variables. Only works for functions of a single variable. So I can choose to simply define this. This is my thickness, dh, which is very small. And that means that the change in pressure here, well, that's dp dh multiplied by dh, right? So dp dh is that expression multiplied by dh. That's going to work. Let's try to do exactly the same thing then, but for gravity. So I'll try to separate those. Gravity was orange. If I had one particle or molecule or something, you know, I hope, that the potential energy of such a particle is the mass of the particle multiplied by the gravitational field constant, multiplied by the height we put it at. But I wasn't directly interested in the potential. I was interested in the pressure. So pressure is force per area. And in my case, the area is going to be unity, because I can always set my scale here so that a equals one. So then I need to get the force. And also I don't want, well, the force on one atom, I'll write that first, the force on one atom, that is minus the derivative of the potential energy, right? So the force on one atom would be minus fg. The reason for that minus sign has to do with the direction of the coordinate axis. The coordinate axis is pointing up, so the force is pointing down. What does this mean then? Well, the pressure difference across the small volume h here, what's that going to be? And in this case, I find it easier to not work with the derivative but the separate components. So then dp, the total difference in pressure, that's going to correspond to the force per volume then. So that's going to be minus mgn, because there are n particles per volume here, right? Multiplied by the thickness dh. Again, one particle minus mg, the force on n particles per volume is minus mgn. And then I multiply it by the thickness of the slice. So the total differential or difference here in pressure across that volume is going to be minus mgn dh. And that I can also write as dp dh equals minus mgn. But now I have two expressions for dp dh here and here. So let's get back and put them in the same equation. So that says that dn dh, multiplied by kt equals minus mgn. This is a differential equation. There are lots of constants here, but n is a function of h, and here are some constants, constants, and then we have the derivative of that function. I actually find this remarkably easy to solve just by writing it in a different way. Divide both sides by kt. So then I have that dn dh equals minus mg divided by ktn. And again, n is a function. So what I'm saying here, I have a function that has the property that the derivative of the function is proportional to the original function, and then we have a constant that shows up in front of it. You know what that function is. That is the exponential function. Actually, it's not just the exponential function. Exponential function, a constant before the exponential function to write. Any constant multiplied by the exponential function will have this property. And this extra term is then that what shows up in the exponent. So based on that, we can write that n equals c, the constant, multiplied by the exponent, raised to minus mg divided by kt, multiplied by the variable h. That was the variable that we had, that n was a function of. And that is a beautiful result, because if you now use the expression for the potential energy, what does this say? Well, this says that n is proportional to, and then I can get rid of my constant, e raised to minus e pot divided by kt. So of course, because it was a special case, it's just the potential energy here, but that's the only energy I had. And that derives the Boltzmann distribution for this particular case. What did I demonstrate? Not really a proof, because it's special. But again, what this says is that if I have very high energy here, up here, then I'm going to have very low number of particles. But if I have very low energy, I'm going to have a very high number of particles. But it doesn't mean that everything will be here, and it doesn't mean that it will be equal. You can, of course, doubt the value of this, because this is only proven for a gas case. It doesn't have any validity directly for proteins, but for now you're going to need to trust me that this will apply universally, and it's a very fundamental result in statistical mechanics.