Let's start by looking at some layman definitions that you can find in dictionaries, online dictionaries nowadays. So this is from dictionary.com, gave two definitions. Number one, the point at which an issue, idea, product, etc., crosses a certain threshold and gains significant momentum, triggered by some minor factor or change, okay. So this is important that you pair these things together, significant momentum, minor factor or change. And then the second definition given by dictionary.com is the point in a situation at which a minor development precipitates a crisis. So a crisis of course, something very big and is a minor development. So this is the thing that perplexes many, many people that you can have very small changes and then very significant outcomes. Okay, now if you look at Merriam-Webster, they define the critical point in a situation, process or system, beyond which significant and often unstoppable effect or change takes place. Here, there's no mention of minor okay but you you like to pair those things up together okay. Now, these are the layman definitions. So what do the scholars say about this? Of course, you can see that both all these things are talking about sudden change. So let's first focus on sudden change, okay. So it turns out that depending on where you come from, that means what discipline you come from, you learn different names for sudden change. So if you're in the Physical Sciences, then you will learn that we call sudden changes, Phase Transitions, or Critical Transitions. So we use them more or less interchangeably. If you come from Ecological Sciences, then you will either call these Regime Shifts or Critical Transitions. And rarely do you find ecologist calling these kinds of sudden changes, Phase Transitions. And if you come from the Socio-economic Sciences, then you will hear them talk about Regime Shifts or Regime Switches. And usually will not talk about our Critical Transitions or Phase Transitions but they actually mean more or less the same thing. But are we sure that they mean more or less the same thing? Since I'm a physicist, so let me take that the vantage point of physics and talk a little bit about what are the general features of a fixed transition. So we start from something that everyone should be familiar with. Unless you don't come from this galaxy, okay? Because we here at least on this planet, we are 70% water. So you're very familiar with the phase diagram of water. So a phase diagram is typically plotted pressure against temperature. And of course the thing here is not to indicate whether there is some function that is changing, how the pressure is changing temperature but whether water exists as ice, okay. So this is a picture of an ice cube or liquid, a glass of water or steam. So these little things here is the actual water droplets. Okay, so the reason why you can see them is because they coalesce in the water vapor and coalesce back into water droplets that scatter light very strongly and therefore you see them. A water vapor of course is invisible to the naked eye. Okay, but we always associate pictures like this we've seen. So we know that water as a very ordinary substance can exist in three different phases, a solid phase, liquid phase and a gaseous phase. Okay, now what happens when we keep the pressure fixed at one atmosphere? So this is one atmosphere, this is way below 1 atmosphere and this is like 218 atmospheres. I don't know what the temperature like in the middle of the earth or on Jupiter, but maybe this is close to it about 218 is something that actually experimentally you can actually accomplish. And what happens is that if we keep the temperature fixed by the pressure fixed at 1 atmosphere and we start varying the temperature, okay, we will find a sudden change from the water being ice and changing very suddenly into liquid water. Okay, so this happens at 0 degrees, it doesn't happen over a range of 10 degrees, so it does not start from minus 5 and ends at 5 degrees Celsius, it only happens at 0 degrees. So this is very sudden as far as temperature is concerned. And of course, if you continue to raise the temperature, then you will find that at 100 degrees Celsius, liquid water evaporates to become water vapors. Okay, again this change is sudden, as far as temperature is concerned. Of course, if you're talking about time, you're watching the kettle boiling it takes actually maybe about two 20, 15 minutes for the water to completely boil off. While this is happening, the temperature remains constant at 100 degrees Celsius, and actually as this is happening, you cannot increase the temperature very much, unless you give it a big shock, okay? So, and more importantly, as we go across the face boundary between ice and water, we have to supply heat, okay. So if you stop this supply of heat actually, the Phase Transition will not be completed. And again as you cross the face boundary between water and steam, you need to again, supply heat, and these two heats are called latent heat or melting latent heats of vaporization. Now why do I bring those up? And that has to do with what physicists call criticality. So, earlier on, you mentioned something about criticality using the Ising model and later on of course generalizing it to a self organized criticality. So let me explain a little bit in more traditional terms what criticality means. Okay, so I mentioned that when you cross a face boundary you need to supply heat that is called latent heat. And actually it turns out that when you have latent heat, then a fiscal quantity called entropy is changing this continuously. Okay, so the entropy of water and, entropy of ice and entropy of water at the same temperature they are different. There's a step, there is a sudden jump as you cross as you go over from water to ice. And because of this continuity, a lot of variables other kind of thermodynamic quantity that you measure will diverge. For example here, okay, what you're seeing here is actually the specific heat, the heat capacity of liquid helium. Okay and actually liquid helium can become a super fluid at a certain temperature and this is the super fluid transition. So, ordinary liquid helium to superfluid helium. The change takes place around this temperature, of course, they have gotten rid of the actual temperature itself is very low. You can only do this liquid helium temperatures, and you can see that you have a very low specific heats and you're far away. Not very high specific heats when you are not that far when you are above the critical temperature, but right at the critical temperature the specific heat actually shoots off, just goes through the roof. Okay, now, there's another feature that is shown in this particular graph. And that is, you can see here that I am measuring in Degrees Celsius, or you can also think of it as measuring Degrees Kelvin, and this range here is about 3000. Now let's zoom in closer to the Critical Transition. Let's scale into 1000 times so this is now measurements in millidegrees. Okay, so this is actually 6 millidegrees above the critical temperature. This is -4 millidegrees below the critical temperature and you look at the specific heat if you plot it out very carefully. So again, you should be amazed at how good these experimentalists are because they can measure all these points so well at this kind of small differences in temperatures. But they know that's really how good the people are, okay? And that's why it's in the best measured physical quantity is actually the I don't remember what it's called, but it is measured to 17 significant figures. So there are a few other things that are measured to the kind of degree of accuracy. This is of course not comparable, but you can see that it is very well done. And you can see that this shape here looks very similar to the one here, which is why this is called a lambda transition because it looks like the Greek letter lambda. Okay, now let's scale in again, let's scale in further, go to micro degrees. Okay, now the data points are fewer, you can see that there are fewer data points and the error bars associated with the data points are also larger because it's not so easy to do clean measurement at this kind of small temperature differences. But you can see that again, the same shape is there. Okay, so here's the question, what kind of function do you know of, when you change the scale the shape doesn't change? If I have let's say an exponential function, I change the scale, that means I changed the way of measuring in terms of degrees, I measure in millidegrees. And the shape will be very different. So instead of exponential growing curve like that, then it becomes flat. Okay, so the only function that we know of, maybe they exist other functions that have this property but at present moment, the only function that we know that has this particular property is a power law. So, this kind of same shape at different scales imply the existence of power laws. Indeed, this increasing specific heat below the critical temperature and this increasing specific heat above the critical temperature, they are both power laws. In fact they don't have to be the same power law, but being power law implies this particular property as well that it is scale free. Because all you have to do when you when you have a power law, you just change the scale. That means if maybe I can write over here. So if I have a power law that is thing something like f (x) = A x to the alpha and I change scale so I let y = k x. So, this is just a scaling rescaling. So, this number can be 100, it can be 10 to the -6, I am just changing how large or small the same thing is. And then when I substitute this back in here, I will find that f (y) = A (kx sorry, I should do it the other way round, right? Yes, I should do it the other way round. So this is x over k okay, alpha, and then you see that no sorry y over k alpha, and you see that I can actually bring out the k. So, this is k alpha and then this is y alpha and I will just call this thing here A prime, y alpha, okay? So, the proportionality factor changes, this number changes, but the function itself does not change, okay. It is still some variable to the power alpha. And this is why power laws are called scale free because when you change scale, the function itself don't seem to change very much, okay. And also what is more important is for power laws that are not increasing, okay? So, in this situation here you can see that both power laws are decreasing as you move away from the critical temperature. So, these are decreasing power laws. So for decreasing power laws, you find that actually they decrease very slowly compared to most other functions that we know of. Let's say we compare them against an exponential. And I have to show I can only show this highlight this difference if I plot it on log scale. So that means the horizontal axis must be logarithmic, the vertical axis must be logarithmic. And then when I plot these three functions, so the power law, the log normal distribution and the exponential distribution. You can see that the power law distribution even though it is decreasing with increasing distance away from the critical temperature, it is decreasing very slowly okay. So this is actually also observed in stock markets. If you look at the returns from stock markets, it is also heavy tail, decreasing very slowly and in fact in the stock market, the power law, the tail of the return distribution actually follows the inverse cubic power law. That means the exponent alpha here is -3 for stock market returns. Okay, so this is how the power law looks like, for the decreasing power law looks like and the alpha actually can be computed. If you're talking about physical systems using the renormalization group theory, lot, you mentioned the renormalization group theory earlier on. But we cannot actually have a lecture on renormalization group theory because that requires you to already have done at least two courses in statistical mechanics. And typically this is taught at graduate level. If you're interested, we can recommend books for you to read up on. Otherwise, just accept that this kind of things can be calculated in general and in of course, you need to assume that there is a scale invariance, in order for this calculation to pull through. And more importantly, a lot you also mentioned earlier on that when you have scale invariance, you can have scaling collapse. Now what a lot of you show is actually one single system. That means we show a sample model, okay at different sizes and then eventually make use of finite size scaling to put all the graphs together and they stacked on top right on top of each other. Now these are data points, these data points here circles, triangles, field circles, open circles. These are data points from different real materials. Okay, and this is actually the scale magnetization. So they have to get rid of the finite size scaling, and also the temperature dependence, implode it as a some kind of a scaled version of the magnetization. And this is a scaled version of the temperature, and you see that all the different materials actually fall onto the same curve. Okay, so I think I've heard Peter and a lot of you mention this, this is called universality. That means it does not matter what the details are, very close to a Critical Transition, everything behaves as if they are the same thing, okay? So this is a important concept of universality, and so I mentioned it here. So, once you can have scaling collapse, for different systems, you can think of these, all the systems as being in the same universality class.