In this lecture, we review topics from SOS 220 (Systems Thinking) that have been presented before the midterm. This includes topics related to introductory systems thinking, causal loop diagrams (CLD's), systems archetypes, resilience thinking, complexity, complex adaptive systems, bifurcation diagrams, tipping points, chaos, basic thermodynamics, Boltzmann entropy, and Shannon information. In the classroom exercise, we mainly answer questions from the students and walk through examples, such as a sample ball-in-a-basin model of a nonlinear system with two alternative stable states (as in a lake that can switch from a clear state to a murky state).
In this lecture, we revisit the fundamentals of the stochastic modeling at the heart of statistical mechanics (microstates, macrostates, multiplicity, and the second law of thermodynamics). We use those fundamentals to motivate the formula for Boltzmann entropy (i.e., why it is a logarithm) and then discuss how systems with relatively low entropy tend to have high "free energy." That is, they have a high ability to do work on another system. When we consider the whole system, the work done by the low-entropy system ends up producing a lot more entropy, causing the entropy of the whole system to increase. This motivates our discussion of the emergence of "dissipative structures" (like life itself) when there are high amounts of free energy (and we use a ball-in-a-basin conceptual model to justify this). We pivot to discussing a combination of economics and physics – econophysics – and how we can think of wealth distributions as macrostates where equal wealth has the lowest entropy and exponentially distributed wealth has the highest entropy. Tidbits of information references are peppered throughout, but we do not get to a specific discussion about Shannon entropy and its relationship to Boltzmann entropy.
This lecture reviews topics from the "Information" chapter by Mitchell (2009, Chapter 3). This chapter reviews the history and fundamentals of thermodynamics and statistical mechanics and how they led to modern information theory. We review topics as in the second law of thermodynamics, Boltzmann entropy, multi-scale modeling, microstates and macrostates (and coarse graining), temperature and other functions of state, Maxwell's demon, and erasure and the Landaurer limit. This lecture discussion sets up a more formal discussion of information, entropy, and free energy in the next lecture period.
In this lecture, we discuss how predictability and repeatability in dynamical systems models cannot be taken for granted, particularly when there is nonlinear feedback combined with either delay or more than 2 state variables. The lecture uses delay in a simple feedback system to motivate how dynamical systems can have SET-based attractors (e.g., oscillations, "omega limit sets"). With set attractors, aspects of the steady-state pattern can depend upon the initial condition. We then discuss how in systems like the Mackey–Glass system (a physiological model), there are ranges of exogenous variables (parameters) that causes the nonlinear dynamics and delay to generate so-called "chaos," an extreme sensitivity to initial conditions where patterns over time do not repeat, and each initial condition is associated with a different pattern. This phenomenon cannot occur in systems of two endogenous/state variables or less unless there is a delay, but it can occur in systems with three endogenous/state variables without delay. We show this for the Lorenz system, a simple model of atmospheric convection. We visualize the "strange attractor" for the Lorenz system using a parametric plot to reveal that the apparently random trajectories of the system truly are structured, and this structure can be revealed by time-series analysis if the right tools are used. Nevertheless, these models show us that for some systems, accurate prediction beyond a short time horizon may always be impossible (even without worrying about stochasticity or bounded rationality) as we will never be able to measure all endogenous-variable initial conditions perfectly.
We begin by rehashing how descriptions of dynamical systems can turn into "ball-in-a-basin" physical models (potential field models) that give us intuition about the behavior over time (BOT) of systems from different initial conditions. This lets us introduce the term "attractor" for the region of a stability regime that "attracts" trajectories from any initial condition within the regime. We use the ball-in-a-basin perspective to compare local resilience (two deep potential wells with a sharp threshold between them) to global resilience (one shallow potential well). The focus on regime shifts lets us discuss both endogenous regime shifts (gentle pushes over thresholds) to exogenous regime shifts (very rare endogenous shocks as well as bifurcations). We refer to the "exogenous regime shifts" as "tipping" and describe bifurcation points as a type of "tipping point." Bifurcation points are thresholds of EXOGENOUS variables that separate regions with different numbers or types of stability regimes. All of these conceptual frameworks then allow us to introduce systems that challenge them -- as in systems with significant delays. Systems with delay are one of the types of systems that exhibit chaos, a term that we will define in more detail in the next lecture.
In this lecture, we start by reviewing resilience concepts in complex systems and how some complex systems can suffer from large regime shifts from one discrete stable state to another. We do an analysis of the equilibria in a simple harvested fishery to motivate the "ball-in-a-basin" potential field perspective on stable states. This discussion allows us to introduce the "bifurcation diagram", which concisely summarizes the relationship between exogenous "slow" variables and the stable states of endogenous "fast" variables; this also allows us to define a "tipping point" as a bifurcation point where the equilibrium structure of the system fundamentally changes.
