Hello, and welcome to the third in our series of lectures on complexity science. Brought to you by the complexity Institute of Nanyang Technological University, I'm Steve Lansing. Today we'll be talking about regime shifts and tipping points. We'll move from the consideration of what causes transitions that may be nonlinear to the ability to forecast such transitions. We'll start by reviewing some layman's definitions of the phrase tipping points. Dictionary.com defines this as the point at which an issue, idea, product etc, crosses a certain threshold. And gained significant momentum triggered by some minor factor or change. Or the point in a situation at which a minor development precipitates a crisis. On the other hand, Merriam Webster defines this as the critical point in a situation, process or system, beyond which a significant and often unstoppable effect or change takes place. In other words, we see the association of tipping point with a sudden change. The phenomenon of critical points and critical transitions has different names in different fields. In the physical sciences, they're called phase transitions or critical transitions. In ecology, we speak of regime shifts or critical transitions and in social and economic science, regime shifts or regime changes. So let's examine the idea of phase transitions in physics, where it's very well understood. And we'll start with the commonplace example of the transitions from liquid water to steam and to ice. This can be captured in a phase transition. At a pressure of one atmosphere, these three phases normally exist at three different temperatures. In a phase diagram, we draw solid curves separating one phase from another. From the phase diagram of water, we see that at a pressure of one atmosphere, we crossed the curve separating ice from water at 0 degrees centigrade. And the curve separating water from water vapor at 100 degrees. These changes are accompanied by the absorption of latent heat of melting and latent heat of vaporization respectively. And because of this, physicists call them first order phase transitions. At lower pressures, it's also possible for ice to transform directly into water vapor via a process called sublimation. This also requires the absorption of a latent heat of sublimation, when we cross the curve separating the solid and gaseous phases of water. These three curves, namely the sublimation curve separating solid from gas, the melting curves separating solid from liquid and the vaporization curves separating liquid from gas. Can all intersect at a triple point where we find the coexistence of solid, liquid and gaseous phases. For water, this triple point occurs at a temperature of 0.01 degrees Celsius and the pressure of 0.06 atmospheres. For most substances, the sublimation curve continues to zero temperature and zero pressure. Whereas the melting curve continues to infinite temperature and pressure. However, the vaporization curve typically ends at a finite temperature and pressure at a critical point, where the phase transition is of second order. For temperatures and pressures above the critical point, the substance exists in a supercritical fluid phase and we no longer have the liquid to gas transition. It turns out that there are amazing physics at and close to, the critical transition. In this figure, what you're seeing is this specific heat of liquid helium. This specific heat of a substance, is the amount of heat energy that you need to apply to raise its temperature by one degree Celsius. Helium is a monatomic gas down to four degrees Kelvin or minus 269 degrees Celsius. At this temperature, it becomes a normal liquid that we can easily store in a vacuum fluid. But when the temperature is lowered to 2.17 degrees Kelvin, at a pressure of one atmosphere. Helium undergoes a critical transition from a normal liquid to a super fluid. A super fluid has zero viscosity and the helium can creep out of the vacuum flask. As we can see from the leftmost graph, as we increase the temperature from below the critical temperature, this specific heat increases dramatically. And after we pass the critical temperature, this specific heat falls off dramatically with a further increase in temperature. Until it starts increasing again, far from the critical temperature because of the shape of this specific heat curve. Physicists call this transition, a lambda transition because it looks like the Greek letter lambda. What's amazing about this lamda transition is. If we rescale the temperature difference from degrees to millidegrees to get the specific heat curve in the middle, the specific heat curve looks very similar at all of these scales. Finally, if we look for the divergence at the critical temperature at different scales. Moving from degrees millidegrees to micro degrees, we see the same shape at each of these scales. For this to occur, the critical transition has to be a power law. When we plot a power law on a linear, linear scale, it's difficult to distinguish it from an exponentially decaying function or a log normal function. The best way to see the difference between these three functional forms is to plop them on a log, log scale. On this scale, we can see that the power log appears as a straight line, while both the exponential in the log normal functions are strongly downwards. For a critical physical system, the critical exponent alpha can be calculated using a physics theory called renormalization group theory. However, the mathematical preparation needed to do this, is just too much to cover in this introductory course, so we'll not go into it here. Interestingly, physicists have found that if you plot this scaled physical properties of different materials, according to their critical temperature, they fall on the same curve. When this can be done for a scaled physical property, we say that there's a scaling collapse in that property. And that the critical behaviors of the different materials belong to the same universality class. Universality is a profound discovery associated with critical transitions. It tells us that for some systems, we can understand their behavior at the critical point. Without knowing the details of the properties of that particular system.
