NOTES

Phase Transitions

by Ricard V. Sole

Bifurcations and Catastrophes

Let’s explore a few key concepts.

We can study the behavior of complex systems in terms of changes in their stability. We kick things off with a discussion of bifurcation, which is a concept attributed to our friend Poincare. Bifurcations are qualitative changes that occur as we make continous changes to a variable. I like the example of shifting from vegetation to desert (think Sahara expansion in Africa). These clear shifts in the system.

Potential functions are what physicists call harmonic functions in math (any real functions with continous second partial derivatives which satisfy Laplace’s equation, which itself desribes situations of equalibrium).

Symmetry Breaking (SB) is “a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system’s fate.” [Wikipedia 11.22.2020]. When there is a continous change to an order parameter in lockstep with changes in a control parameter, we call such SB phenomena second order phase transitions, but if there is a dramatic shift over change in a control parameter we call them catastrophes. Water to ice transitions at sea level would be an example.

Hysteresis is the dependence of the state of a system on its history.

As the critical point is approached, “the characteristic time needed to reach the equilibrium state rapidly grows”. Noting these changes can give indication that a critical transition is close at hand. This is one of the key reasons I believe phase transitions should be studied - anticipting imminent change.

The author reminds us that in the real world, multiple bifurcations cascade around one another, yielding that hard to model thing we call life.

Percolation

What is a system in the first place? What makes something a part of the system vs. not? This chapter kicks off with this core clarification, concluding that a system is a set of items in which there is a path connecting any two elements.

I had never before considered that for something to percolate through a system, there needs to be a connected path. My mind anchors on the air bubble that percolates up through a liquid. There is a continuous path that the bubble follows. We can swap in the example the author gives of the burning forest, with fire spreading from burning trees to their closest neighbors. For a fire to spread from one side of the forest to the other there needs to be a connected path of adjascent trees close enough to catch embers.

Bethe lattices provide a model of percolation that grants us the ability to calculate a percolation probability.