Phase Transitions & Bifurcations

A phase transition is the transformation of a system from one state to another through a period of rapid change. The classical example of this is the transition between solid, liquid and gaseous states that water passes through given some change in temperature, phase transitions are another hallmark of nonlinear systems. In this module we discuss the concept in tandem with its counterpart bifurcation theory.

Bifurcations & Phase transitions
As we have previously discussed 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 an 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 diver of what we call phase transitions.
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 systems 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 changes 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.
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 parameter can causes 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.

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[JoseAfonso]

This whole series is absolutely awesome. Excellent production and great content :)

[aakashguglani]

I loved this series. I think Eastern philosophy especially Advaita Vedanta in Hinduism and Madhyamika Buddhist doctrine is based upon the non-linear system. It is important to understand the initial position that leads to these positions. Or the Butterfly effect. Loved it. Gave me a new topic to research.

[Kellerwesselklaus]

You are amazing! I love your videos.
I'm a wannabe writer, and the theory emergence, and nonlinearity inspired me. I am trying to write a nonlienar story right now, where the overall plot emerges from the synergies between the local plots, and this story is about a phase transition of a man, a family, and a society (in a fractalic fashion).
Sorry for my unreadable english, I just wanted to say, that your work is awesome, and make more, and we love you

[Zayden.]

Punctuated equilibrium is seen in human societal development over the last 100, 000 years. From hunter gatherer society to agricultural society. The rise and fall of the roman empire. The transformation in europe from feudalism to capitalism, which spread throughout the world. And now we are living through world capitalism's convulsions as it reaches punctuation from equilibrium, going to either world socialism with planned economy with reduced workweek or catastrophic global warlordism. Socialism or barbarism as Rosa Luxemburg said.

[martincremer]

Very nice work! The writing has a clear and concise tone considering the relative 'complexity' of the topics. The voiceover is soothing and plays well with the cool pallette of colors. The subtle animation of diagrams perfectly complements communication without sensory overload. I'm curious as to whether this is the work of one person? Thank you for the thoughtful articulation of these important constructs.

[janderson]

I would argue that you can indeed talk about the viscosity of gas

[joshdominguez]

So this unexpected COVID situation can be represented by this punctuated equilibrium, where there's a transition going from negative feedback stable state to positive feedback unstable state/equilibrium?

[idiosinkrazijske.rutine]

Yes you can talk about viscosity of gas.

[SuperCycleBear]

Another example of bifurcation in action is the emergence of cryptocurrencies. Choice of which path you stay, fiat or cryptos, could define your long term future.

[qianlin]

Thanks for your amazing video! Extremely clear at explanation and vivid examples. I was wondering about the example curve for Punctuated equilibrium, where the whole process consists of negative, postivie and then negative feedback. Is there a simple differential equation to characterize such process?