 So these curves of how entropy varies as a function of energy are going to be really instructive to understand changes. So let's look at some. Let's start with a simple case we had, when as a function of energy we have the entropy increasing gradually. We're going from a state 1 to a state 2. That means that there is a temperature in state 1 and there is a temperature in state 2. Just based on that shape, we can actually reason roughly how the energy should change as a function of temperature instead. So if we move from temperature 1 to temperature 2, the energy would maybe go up something like that, nothing special. But then the question is if we stop halfway here, what is how many systems or molecules whatever are going to be in state 1 and state 2? So that somehow as a function of energy going from 1 to 2, we can have some weights here. Let's just consider three alternatives. First, if I'm at the temperature corresponding to energy 1, I'm going to be entirely in state 1. Or at state temperature corresponding to state 2, I'm going to be entirely in state 2. That's kind of obvious. In this case, there is absolutely nothing special happening here. I can be stable at any point here. So if the system has an energy corresponding to the halfway point here, that will correspond to the system on average being halfway between the two states. This is an important change that we see and it corresponds to boiling water raising the temperature to say from 50 to 51 degrees centigrade. Pretty boring but important. Of course, I can stop the boiling process and leave the water at 50.5 degrees centigrade and then the molecule will have that temperature on average. But we're not going to fold any proteins that way. So let's look at something more interesting. Remember that we talked about the energy landscapes and these peaks and that the peaks corresponded standing on the edge of a knife and that that corresponded to a curve having the opposite shape. So let's imagine such a process. What happens if we need to go over a barrier? Then we're going to have the same entropy versus energy and then there's going to be something that goes up maybe like this. There will be a temperature 1 here, there will be a temperature 2 there and then halfway between them it's a bit difficult to say, right? There might be some sort of average temperature if I just draw a line between those two states. It turns out as I'm changing the temperature from temperature 1 to temperature 2, there's actually nothing particularly extreme happening to the energy. The energy of the system will go up that way smoothly. So I can definitely be at an energy that's halfway here, I can be here. Or can I? Well, I can for the system. But if I look at the population of states for individual molecules or so, weights, at state 1 and state 2, they're going to be the same as before. So at state 2 I'm there and at state 1 I'm there. But if I stop halfway through here, well, I can balance something at the edge of a knife, but at room temperature and equilibrium, at finite temperature, things will always be hitting me. So what's going to happen is that half the molecules will fall down to the left and half the molecules are going to fall down to the right. So this will actually lead to a population being half at state 1 and half at state 2, but nothing halfway between them. And that corresponds to this part of the face diagram or the end of the kind of being not disallowed but unfavorable. I can balance here, but I will not be stable here for a long time. And you will not find tons of system balancing on the edge of that knife. If we formulate that in terms of how likely it is to observe the system here, the system has to pass through this barrier to get there, but the likelihood of finding the system there at equilibrium is actually zero. It will always fall down to one of the other two states. That is a so-called all or non-transition and a phase transition and corresponds to boiling water from liquid water to form water vapor or melting ice to form liquid water. But you might say, that sounds strange that that's happening over an interval. You all know that there is a specific melting and boiling temperature. Well, that's actually not true. The book Finkelstein goes through some effort to show that you can calculate the temperature span here, delta T, and show that delta T corresponds to roughly, he has to make a few assumptions and approximations, but it's roughly 4K and then this intermediate temperature where the transition is happening, divided by delta E, that the difference in energy between the two state surfaces. And here's the important thing. If I'm melting ice, I'm not considering one water molecule. You're considering a cube of ice. And if that is, say, a kilo just to make the math easier, the energy levels we're talking about are K-cals. Note K-cals, not per mole K-cals. And that means that the temperature interval is going to be the Boltzmann constant with something tiny divided by something that is reasonably large. That's going to be a tiny temperature interval. So, you know what? There is actually an interval at which ice is melting to water and that might be from 0 degrees to 10 to the minus 23 degrees. That's an extremely small interval and I think you will agree that it's pretty decent to approximate that with 0 exactly, right? Can we do something else with this? Well, let's consider a protein instead because there will be parts of protein folding that will behave this way. But I'm not folding a kilo of protein. I'm looking at one protein molecule. And in case I'm looking at one protein molecule, 4K, well, T is going to be the same ballpark, but the energies we now talk about are K-cals per mole. And that per mole thing are going to cancel the Boltzmann constant, roughly, right? So, for proteins, you will have the same type of transitions, but they will happen over a span of several Kelvin, maybe 5 or 10 Kelvin. And in that span, you will have that it's either stable here or stable here, but you will never be able to capture the system halfway through. And that leads to some interesting considerations about stability, where the stability might occasionally depend on how fast the process has happened.