In previous posts, we have discussed the theory of phase transitions, and how these theoretical concepts can be applied to understand the dynamics of complex systems including earthquakes and markets. An important idea was the concept of metastability, the temporary state of a system that is different from the long-term, preferred state of the system.
The idea of hysteresis is associated with metastability, but an important question relates to the process by which the system evolves into a metastable state. To answer this question, we return to the Landau Theory of phase transitions described previously.
We defined the order parameter of the system, together with a cost or fitness function. The order parameter describes the current state of the system, while the cost function defines the dynamics by which the system evolves. The system is driven by a persistent forcing f. For an earthquake fault, f is the plate tectonic forces.
For markets, f is the liquidity or money supply. For earthquakes, convective circulations deep within the earth’s mantle produce the tectonic forces that drive the plates. For markets, the federal reserve and other central banks supply the money necessary to grow the economy.
The Landau Theory postulates that the system evolves towards a state of minimum energy. This state can either be a local minimum, or a global minimum. An example for an earthquake fault is shown in the figure below , where there are two local minima. In this case, the order parameter is the deficit in slip on the earthquake fault.