So far I've given you some historical background on phase transitions as studied by physicists. I said that, renormalization group theory is too complicated we can't talk about those, but is there a simple physical theory of phase transitions? It turns out that there is and it is called a Landau theory. Now of course left Landau is a contemporary of Richard Fineman. He also won a Nobel Prize in Physics. So this really smart guy, okay, but he came up with a very simple theory. Okay, so his idea is that first there exists a lot of free energy. Don't ask him why there's no such thing must exist but he knows that it must. Okay, and if you ask him that this kind of questions in his lecture, I'm not sure what he will do to you, but he will write down something like this. So, you can see that it is not very difficult to write down. There is a function F Landau free energy, it is a function of a variable called phi, it is actually called the order parameter. And of course if you can think of it doing a Taylor series expansion about phi goes to zero and this is what you will get. Okay, of course it should be a linear term and then it should be a cubic term. For him what he's interested in is your move first, move yourself to the minimum position before you expand, so that you don't have the linear term. And then the cubic term actually, he got rid of by symmetry arguments. He's saying that left is not different from the right and therefore you should not have a cubic term and this is one of the simplest Landau free energy that you can write down. So let me show you what this Landau free energy looks like, okay? Now notice that is a parameter t here that you can change, there's nothing else you can change, the phi is a variable, so you must must plot the function as a function of phi but you can change the value of t. Which is actually the scale temperature in a physics interpretation. So if t is greater than zero, then what happens is that you are adding a positive quadratic function to a positive quartic function, so the only shape you can get is just this bell like shape. Okay, this this bell like shape and so here's the key thing the key idea behind Landau theory. So, this is the order parameter of the physical system but this thing here does not occur in real life. In real life you only find the system at the minimum point of the Landau free energy. So, where is the minimum of the Landau free energy 40 euro? It is at this point, right? Okay, and there's only a minimum, so that means, you you normally would not find the system here, you won't find it here, you only find it here, okay? So this is the minimum position, so I marked the minimum position of this Landau free energy in red okay? Then I go on to here. Okay, this is when the scale temperature is zero, that means the quadratic term disappears and there's only the quartic term remaining. The quartic term has this very flat shape, so that's what this thing is indicating, but the minimum position is still at phi equal to zero, so I plot it down here. But once you get below t less than zero, so when the skill temperature is below t equals zero, which turns out to be the critical temperature, then you can have a downward flexing quadratic function competing with a upward flexing quartic function. The quartic function will win because it has a much stronger power. Okay, for highly positive phi, this quartic term will win. But in between no for intermediate values of phi, it turns out that you can have this kind of shape down here. So this is called a double well potential okay, now the minimum Landau free energy is no longer at phi equal zero, it occurs at plus and minus of some number that you can calculate from here. So all you have to do is actually take the derivative of this function okay, set the derivative to zero and then solve for the values of phi for which that is satisfied and you end up with three points. Of course, you end up with zero a no a plus minus some phi not, some values of phi that is actually a square root function of the absolute value of the skill temperature t. So I have plotted actually showing you how the Landau free energy like for different values of the scale temperature. And if I plot the fixed point as a function of t so this axis is t, you can see that I go from one fixed point to two fixed point as I cross t equals zero, okay in mathematics this is called a bifurcation okay? So, you go from one fixed point to fixed point, so this is bifurcation down here. And in fact this is called a pitchfork bifurcation, phi equals zero remains a fixed point that means if you start the system off at phi equals zero, it won't move anywhere because the force acting on it is zero. But if you will displace it a little bit it, it will go off to either one or the other minimum points and therefore this line here becomes unstable I didn't draw it. So, sometimes in textbooks you will see people drawing dash dash dash to indicate that is unstable solid lines means that is stable. So, for t less than zero the stable fixed points are along this parabola here, okay? So this is called bifurcation diagram and actually this theory here describes second order phase transition. So, I earlier on I did not mention after talking about the phase diagram water, what are the different kind of phase transitions that you can find. It turns out that in thermal physics, there are only two kinds phase transition that we observe so far. There might be more I mean in principle there can be, but we have not seen anything beyond a first order phase transition and a second order phase transition. And this phase transition is also called a continuous phase transition because this change here is continuous, you can see that the fixed point changes continuously but not differentiable. At this particular point now here and in fields outside of physics, they will say that this is not a regime shift, this is just a continuous change, although it does not have a finite derivative at this particular point here. Now, would Landau theory also be able to explain first order phase transition? And it turns out that yes, it is possible, but you have to add a linear term to the Landau free energy, okay? So everything else is the same except that you add one term and of course we know this theory is only interesting for t less than zero. So it's not so interesting, if it is here, so I add a linear term I change the linear term, nothing much will happen actually there will be one fixed point that will keep on shifting about. So we do this for fixed points when t is less than zero. Okay, so you see the little h down here and the phi down here. So, if you think about the Ising model, if you think of phi as the magnetization of an Ising model that you have shown earlier on, then this natural interpretation for h is actually the magnetic field. Okay, so you think of how you have an Ising model that has two possible phases, a phase where the magnetized or the North poles are pointing up versus or the South Poles are pointing up. And then h can change that, you can switch the magnetization around by playing with h and then again we know we plot the Landau free energy, so we can go from something like this to something like this. So as you change the value of h, you can see that the tilt of the two wells go from something like this to something like that and this is h equals zero, the same situation as here. And if you draw the fixed points, you see that there are two branches and then in between of course, they are connected by a line of unstable fixed points. And so these are the fixed points and this describes the first order phase transition, how does it do so, what happens is when you go from positive slope to negative slope, okay? So you are following basically this line here, so if you are in decreasing your h, you'll find that when h reaches zero there's no problem being here because this is a local minimum of the Landau free energy. [INAUDIBLE] Here either because this is still locally minimum. But once you get to this point here, okay, this first well here disappears because you tilt the system so much this first well disappear the second wells become extremely deep and you roll suddenly over to the right well. And then after that when you increase your h backwards you find that when you reach this point here there's no problem, the system can be here when you reach this point the system can be here. And you actually need to tilt the system quite a lot for this particular well to disappear before the system goes back there. This particular phenomenon where you enter phase one at this value of the drive the control parameter, and then when you come back the other direction you return to phase one at a different value of the control parameter. This phenomenon is known as hysteresis. So it's very common for magnets and it's a hallmark signature of face first order phase transitions and everyone agrees that this is a regime shift. So when economics talk about regime shift people in ecology talks regime shift, they are mostly talking about first order phase transition, okay?
