Self Organization, Emergence, and Stable States in Complex Systems

One of the hallmarks of complex systems is self organization. What this means is if we leave a complex system alone, and let it evolve over time on its own, we will find structures appear on their own, without guidance from an external agent. These structures are said to be emergent, because they do not seem to be encoded in the interactions between different microscopic parts of the complex system. A commonly cited example of emergent structures is the anthill (~1 m in size), assembled by ants that are very much smaller (~1 cm) in size. We know the ants could not have an anthill design stored in their 250,000-neuron brain, because every anthill is unique, even though they share many common functionalities. Another common example of emergent phenomena is the murmuration of starlings. Starlings are quite a lot smarter than ants, but because murmurations are dynamic, it cannot be the result of a central authority. In fact, starting in 1986 Craig Reynolds and others demonstrated using computer simulations of boids that murmurations can be explained by a few simple rules between interacting neighbours.

Emergent structures change in space and in time, but they are surprisingly long-lived. Therefore, they are also referred to as stable states or stable regimes. A complex system typically has more than one stable state, and is thus said to be multi-stable. Stable states are qualitatively and quantitatively different from each other. A simple way to think of them is as valleys in a landscape. If the system is close to the lowest point in the valley, its behaviour would be qualitatively similar to the one at the lowest point. This would be the case until we bring the complex system across a ridge that separates the initial stable state from the final stable state that we will move into. We think therefore of stable states as basins of attraction, separated by ridges. This information is best presented as a phase diagram.

In Figure 1, I show a schematic figure of the phase diagram of water and the three phases we are familiar with. In this figure, we see a solid curve separating the solid phase from the gas phase, another solid curve separating the solid phase from the liquid phase, and yet another solid curve separating the liquid phase from the gas phase. This last curve does not go on forever, but ends at a critical point (pC, TC). Beyond this critical point, liquid water cannot be distinguished from gaseous water, and water is said to form a supercritical fluid. There is another special point on this phase diagram, (pTP, TTP), called the tricritical point, where the three solid curves meet.

Many others before me have recognised the utility of presenting stable states of a complex system in the form of a phase diagram. For example, many have suspected the stock market to consists of multiple stable states (see the recent work by Munnix et al. in 2012, https://www.nature.com/articles/srep00644), and suggested that they be organised into a phase diagram (see the work by Kutner in 2002, https://www.sciencedirect.com/science/article/abs/pii/S0378437102010580). In fact, in the 2018 book chapter contributed by McKelvey and Yalamova to Governance and Control of Financial Systems, the authors showed a schematic phase diagram of the stock market that looked very much like the phase diagram of water shown in Figure 1.

For a system of pure water, the way to change its phase is to change the temperature at a fixed pressure (or change the pressure at a fixed temperature). At one atmosphere of pressure, water exists as ice at temperatures below 0 degree Celsius. When we increase the temperature, water remains as ice, until we reach 0 degree Celsius. At this melting temperature, ice starts to melt to become liquid water. So long as there is ice remaining, the temperature will not change. Therefore, we can think of this solid curve separating ice from liquid water as a point on the phase diagram where solid and liquid coexists.

Once there is no ice left, and water is entirely in the form of a liquid, the heat we supply into the system will start increasing the temperature of the water, until we reach 100 degrees Celsius. At this temperature, liquid water begins to boil and become steam. Again, as long as there is liquid water remaining, the temperature remains constant and the heat we supply to the system is used entirely to convert liquid to gas. Therefore, the solid curve separating liquid water and steam as a point on the phase diagram where liquid and gas coexists.

In the talk that Prof Didier Sornette recently gave to CERN, one of his slides is on the hypothetical 'management' of liquid water close to the boiling point. If we are not aware that water boils to become steam at the boiling point, with a tremendous increase in volume, and keep the water confined to a small sealed container, we would end up with a catastrophic explosion.

In the next article, I will explain what happens when we tune a control parameter to continuously bring a complex system across a phase transition (also known in other disciplines as a critical transition or regime shift). The mathematics involved is known as bifurcation theory or catastrophe theory.