 This is not very hard to derive. Surprisingly. I know that I might have hinted that it would be, but I wanted you to have a gut feeling what we're aiming for before we try to derive it. You might find this hard, but in that case, what's this concept two or three times? The math here is not hard, but I think what some of you find hard is that we don't really have anything to start from. We have to define things from scratch and then derive something based on our definitions. But you know what? You're allowed to do any definitions and when I do definitions, I occasionally go wrong. I can't go wrong here, but rather I could, but it would be stupid in terms of didactics. But don't worry if you go wrong. Spend an hour on it and throw the paper and start over. But I'm not going to be throwing things away. I'm going to start by defining something. And as a physicist, if I want to think of something, we need to define a system, right? So I'm going to start by defining a large system. And this large system could be the universe or something. And I'm going to say that the total energy in my large system is E0, which for this system is constant. It doesn't change. But then I will look at the small system where I say that I have an energy epsilon. Now, what I'm interested in is how likely is it for this small system to have the energy epsilon? And the small system might be, for instance, how likely is it for the pen to be in this confirmation? Now, that means that the rest of the system here will have the energy E0 minus epsilon. And if this is a very small part of the system and if epsilon is a small number, that means that there is a lot of energy here. And that means that there are lots of ways that you can distribute this energy, right? That means many states. On the other hand, if epsilon is a very high energy, that means there are going to be less energy here and fewer ways. And I already hinted that it's going to make it very clear now that the likelihood of observing a system in a particular state, well, that's simple, is just proportional to the number of ways I can find the system in that state. Compare that to the previous slide in orange, right? If it's a well-ordered system, there are few ways we can put it that way while there are many disordered ways. So, this is the first part. I'm going to need to define something. And I said that I'm interested in the probability of observing a system. And that probability P is proportional to the number of states. I'm going to need to call that something. I would follow the book and call that M. We typically call this a thermostat or something, but that's the only system I'm going to be considering. So, I will skip the suffix here to save some space on the board. But the number of states here, in general, is a function of the energy. And the energy for the rest of the system here is E0 minus epsilon. Because, again, this is the small part of the system that I'm not considering. That is well defined. That's a specific state, right? It's these other states that I'm allowed to distribute and consider how many of them I have. I don't know anything more about that, literally. Or do I? Well, I know that it's a positive number because there should be at least one state. The other thing with probabilities is that they're a bit of a pain to work with. If I have one coin here that's head or tails and one coin here that's head or tails, they have two possibilities each. But if I start grouping coins and I talk about probabilities, that means I should multiply probabilities. That's a bit awkward because when I talk about energies, we're used to summing them. There is a very easy way to get the same property here. I will just introduce another number or whatever we call it here. S. Don't think about this. I could have used z if I wanted to. S equals, leaves some space, logarithm of the number m of e0 minus epsilon. And then I might want to have some units here. So I'll just say that there is a small unit that I would call kappa. Just a constant. I'm always allowed to take the logarithm of something that's neither 0 nor negative. So that I haven't simplified or assumed anything here. It's just a pure mathematical definition. But now if I double, if I take two systems like this, I should double the energy and I should double s. So that makes them easy and behave in similar ways. But I still don't know. I didn't know much about m. I don't know a whole lot more about s. Remember when we defined bonds and what a physicist did when there was a complicated function I didn't really know the shape of. I made a series expansion around something that I knew. Let's try that here. So let's do a series expansion of s and say that s is approximately equal to, but I'm going to write equal for now. So let's expand this around the value e0. So s is then equals s at the value e0 plus the displacement from that value multiplied by the derivative, right? That's the first order component. But the displacement I'm going to looking at is what I'm adding here and I'm not adding. I'm actually subtracting. So it's going to be minus epsilon. So it's minus epsilon multiplied by the derivative of s, which is a function of e, with respect to energy taken at the value e0. Just first order expansion, Taylor expansion. And now I'm going to need to continue this with a second and third and fourth order term. Or should I just make do here and say that we're good enough? Yes. And here we're going to use a trick. Remember that I said that s was proportional to the size of the system. If I double the size of the system I will double s. That means that that term will go up as the size of the system. So I definitely need that one. Here epsilon is small, but here I have a quotient of something in the denominator that is proportional to the size of the system and something in the denominator that's also proportional to the size of the system. That's going to be a constant term. I need that. But if I were to take a second derivative here, I would have something in the denominator that is proportional to the size of the system divided by the square of the size of the system. And the trick with statistical mechanics is that we study very large systems, ballpark of 10 to the power of 23 atoms, right? If you divide one divided by 10 to the power of 23, it's going to be so large that we can literally ignore anything beyond this first order term. And it's not really a simplification for a system that's large enough. So now I have a much simpler expression for this. Let me call this something let's say that the first one here was equation one and the second one was equation two. I'm now going to solve for m because I really wanted m right. I'm going to solve for m in equation one, but instead of saying s, I can use equation two. So where will that lead us? Well, this is easy. I'll take s divided by kappa and then take the exponential. So that m of e zero minus epsilon is equal to e raised to s, which again is a function then of e zero minus epsilon divided by kappa. That's just the plain definition. So let's then insert equation two here. That corresponds to the exponential first term s at the value e zero divided by kappa minus, well, the addition here corresponds to multiplication, right? So I can make this a separate exponential and then have a minus sign up here. E raised to minus epsilon and then I'm going to need that entire expression. ds de taken at the value e zero divided by kappa. And this might not look simple, but you're done. This is amazing. s of e zero. The whole point here is that e zero is constant when I look at the world. So this is going to be one big fat constant factor. And we're talking about proportionality. So who cares about a constant? We can strike that out. Here, things look more complicated though. You could argue that if we were defining physics, we might say that the energy here somehow, that depends on how much energy I have in the small system. So that's definitely something that I have to consider. While this whole other part has to do with derivatives and connections, how entropy is changing relative to energy. So somehow the second term here describes, as I'm exchanging energy here, how would that influence the number of states? I would likely have called that x if I saw this the first time. But the point is that we're not the first ones to see this. So if you do that math, leaves, actually I'm going to move down here. It turns out that d s d e corresponds to 1 over t. Instead of x, I can say 1 over t. It's just a number for now. And instead of kappa, let's put k there. Let's call kappa. That seems to be a fundamental constant. That will mean that this m equals a constant times e raised to minus epsilon divided by kt. And that is the proportionality you were looking for, right? We've redefined both, re-derived the Boltzmann distribution. But the cool thing, we have now derived the Boltzmann distribution without really assuming anything. This can be any system. And remember, the entropy came out naturally here. We never really defined entropy. We were just counting states. And then it turned out to choose to be convenient to introduce the logarithm of the number of states. What that then gets us is that we have this simple equation with two parts. The epsilon part here, that is just how much energy I was putting in my small system. Kappa, that's just a fundamental constant. Had we defined this today, it would likely have put in one and have measured temperature in units of energy. But this t thing that you thought that you had a gut feeling for that. You know what? I don't think you have a gut feeling for temperature. What you think is temperature is heat. Because if you're putting your hand on a boiling piece of water, it's going to be warm, right? But if your hand is hit by a water molecule that has say a velocity of 1000 meters per second, technically that molecule has a very high temperature. But you're not going to feel it, because it's just one single molecule. So it doesn't really heat your hand. So you have a gut feeling for heat, not temperature, which is incidentally why we can walk on cold. Don't worry too much about this. This is t. It is a constant that describes the relation when as I'm adding more energy to a system, how would that influence the number of states that I can derive there? To me, I have a much better gut feeling for entropy than I have for temperature. So sorry for ruining that for you. But at least you understand entropy now, if not temperature.