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Phase Transitions

The metamorphosis of a caterpillar into a butterfly is an example of a phase transition

The metamorphosis of a caterpillar into a butterfly is an example of a phase transition

Phase transitions and bifurcations are periods of qualitative and often rapid change in the dynamics of a nonlinear system’s state.1 They involve positive feedback loops that drive the system far-from-equilibrium and result in exponential change and a pattern of development called punctuated equilibrium where periods of stability are punctuated by phases of rapid change called the phase transition period.

Feedback Loops

The qualitative dynamic behavior of nonlinear systems is largely defined by the positive and negative feedback loops that regulate their development, with negative feedback working to dampen down or constrain change to a linear progression, while positive feedback works to amplify change typically in a super-linear fashion. As opposed to negative feedback where we get a gradual and often stable development over a prolonged period of time – what we might call a normal or equilibrium state of development – positive feedback is characteristic of a system in a state of nonequilibrium. Positive feedback development is fundamentally unsustainable because all systems, in reality, exist in an environment that will ultimately place a limit on this grown. From this we can see how the exponential grow enabled by positive feedback loops is what we might say special. It can only exist for a relatively brief period of time. When we look around us, we see the vast majority of things are in a stable configuration constrained by some negative feedback loop, whether this is the law of gravity, predator-prey dynamics or the economic laws of having to get out of bed and go to work every day. These special periods of positive feedback development are characteristic and a key driver of what we call phase transitions.

Transition

Image depicting two different phases to the substance water, the transition from ice to water is an example of a phase transition

Image depicting two different phases to the substance water, the transition from ice to water is an example of a phase transition

A phase transition may be defined as some smooth, small change in a quantitative input variable that results in a qualitative change in the system’s state. The transition of ice to steam is one example of a phase transition. At some critical temperature, a small change in the system’s input temperature value results in a systemic change in the substance after which it is governed by a new set of parameters and properties. For example, we can talk about cracking ice but not water, or we can talk about the viscosity of a liquid but not a gas, as these are in different phases under different physical regimes. And thus, we describe them with respect to different parameters. Another example of a phase transition may be the changes within a colony of bacteria, that when we change the heat and nutrient input to the system we change the local interactions between the bacteria and get a new emergent structure to the colony. Although this change in input value may only be a linear progression, it resulted in a qualitatively different pattern emerging on the macro-level of the colony. It is not simply that a new order or structure has emerged but the actual rules that govern the system change. And thus, we use the word regime and talk about it as a regime shift, as some small change in a parameter that affected the system on the local level leads to different emergent structures that then feedback to define a different regime that the elements now have to operate under.

Examples

Examples of phase transitions might include the fall of the Berlin Wall, before this rapid critical phase transition the global political environment was largely defined by a bipolar regime. Before the fall this bipolar model was the parameter we used to define the system, after the event, the political environment was described with reference to a new set of parameters relating to globalization. The Arab Spring might be another example. The Arab Spring is widely believed to have been instigated by dissatisfaction with the rule of local governments. After many decades of the Middle East being held within a particular configuration or political regime, the Arab Spring was a punctuation of this equilibrium. The previous regime was a set of negative feedback loops that balanced the system into some equilibrium – we might say there was some balance of power – but this balance got broken through some small fluctuation, the self-sacrifice of a street vendor in Tunisia, this small event then got amplified by positive feedback into a large systemic transformation. Through this positive feedback, the balance of power was broken temporarily and the political system across the Middle East moved into a phase transition.

Bifurcation Theory

The logistics map with bifurcation points where a small change in the input value results in a topicality change (the bifurcation)

The logistic map with bifurcation points where a small change in the input value results in a topicality change (the bifurcation)

Another way of talking about this is in the language of bifurcation theory. Whereas with phase transitions we are talking about qualitative changes in the properties of the system, bifurcation theory really talks about how a small change in a parameter can cause a topological change in a system’s environment, resulting in new attractor states emerging. A bifurcation means a branching. In this case, we are talking about a point where the future trajectory of an element in the system divides or branches out as new attractor states emerge. From this critical point, it can go in two different trajectories which are the product of these attractors. Each branch represents a trajectory into a new basin of attraction with a new regime and equilibrium. To take a real world example of a bifurcation, say you have been studying Fine Art as an undergraduate. This subject has for the past few years represented your basin of attraction, that is to say, your studies have cycled through its many different domains but never moved off into another totally different subject. But now that you have graduated, you have the option to continue your studies in either sculpture or painting. You have now reached a bifurcation point as two new attractors have opened up in front of you. Some small event at this point could define your long-term trajectory into one of these two different basins of attraction.

Punctuated Equilibrium

As opposed to linear systems that may develop in an overall incremental fashion, the exponential growth that nonlinear systems are capable of -through feedback loops and phase transitions – leads to a different overall pattern to their development, what we might call punctuated equilibrium. Within this model of punctuated equilibrium, the development of a nonlinear system is marked by a dynamic between positive and negative feedback, with negative feedback holding the system within a basin of attraction that represents periods of stable development. These stable periods are then punctuated by periods of positive feedback which takes the system far from its equilibrium and into a phase transition as the fundamental topology of its attractor states change and bifurcate. Examples of this punctuated equilibrium might be the development of economies that go through periods of stable growth then rapid change through an economic crisis and recovery, or ecosystems as they collapse due to some environmental change and then an ensuing period of rapid re-growth towards a new equilibrium of stable development again. The same punctuated development may be seen within the development of a human being as they go from childhood to adulthood to old age, each period representing a stable basin of attraction with changes between each being marked by periods of rapid and defining change.

Cite this article as: Joss Colchester, "Phase Transitions," in Complexity Labs, May 5, 2015, http://complexitylabs.io/phase-transitions/.
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2017-05-18T15:57:08+00:00