Mathematics, Modeling, Mechanisms (and Movies)
Phase diagram of mathematical models
For a description of this phase diagram read the last section of this page
Increasingly so, the premise of some the most popular movies are based on scientific "fact" or occurence, scientific figures (less so), or some far fetched idea. By way of example, the Edge of Tomorrow is a perfect example of chaos in our lives and how slight changes in intial conditions can lead to drastically different outcomes, 2012 a disaster movie based on the idea of tipping points in Earth's climate, recovering Matt Damon from Mars in The Martian is based on the idea of slingshotting around Earth, drastic mutations that lead to superhero like qualitites in Xmen, using mathematics to build a baseball team etc. etc. The list is endless. Each of these movies can be a launching point for a case study of engaging your curiosity, scientific understanding and mathematical modeling. The Big Short can be used as a catalyst to learn about financial mathematics, I am Legend can be used as a catalyst to learn about the SIR model and the spread of infectious diseases, Lincoln has fascinating examples of voter dynamics ... the list is endless if only you are willing to be curious.
The point of this class is not to teach you lots of mathematics, this is what other classes do, what I want to get across is the ways in which mathematics can be used to ask questions, infer causality, and help us in the understanding the world. In short, to understand mechanisms. Along the way you will learn about a few different phenomena, but the overarching goal is to understand the phase diagram of models as you will see on the course webpage.
The primary deliverable in this course will be a final project, which the students will work on throughout the quarter. Pick a movie that you enjoy, that has some grounding (or not) in scientific fact, start to dig into the basic concepts, the fundamental ideas, investigate the kinds of models out there to understand the phenomena and start digging! Perhaps you will do something original, perhaps you will do a literature review of the material out there. The point is to learn something new about the world, think about how you might go about analyzing it mathematically, and finally...did that movie make any sense, or could your children really be mutants with the power to read minds!
Course outline (with reference to numbering scheme in the phase diagram above)
1. The Harmonic Oscillator (one equation to rule them all)
Learning outcomes: One degrees of freedom (dof) in the presence of no interactions and no fluctuations can still exhibit interesting and non-trivial properties. Simple ODEs, Newtonian mechanics, conservation of energy and dissipation, resonance and blow up (watch the video on this page for a beautiful demonstration of the differential resonance of building with different heights when there is an earthquake). To see things from a different point of view (transcript and documentary).
Related movies and possible project ideas: Ray (the mathematics of resonance and how the vocal tract (and some musical instruments) works), Contact (the mathematics of resonance and how radios work), James Bond (the mathematics of resonance and how lasers work),
San Andreas (the mathematics of Earthquakes)
Learning outcomes: Single degree of freedom, in absence of any stochasticity, can display non-trivial dependences on initial conditions and parameters. Emergence of a probabilistic description from an underlying deterministic process. Angular motion and Newtonian mechanics, universal properties of chaos (Malthusian growth model, logistic map) basic probability
HW: Other than questions in class here are two additional questions.
Related movies and possible project ideas: Possible movie based final projects: 2012 (the mathematics of climate modeling and chaos), The Martian (the mathematics of spaceflight and trajectories). The three-body problem (the mathematics of the n-body problem)
HW1: Assignement (has 3 parts, conceptual, mathematical, and numerical, related to the SHO, coin-tossing, and chaos. The HW isn't easy or short, so I advise you to make an early start). Folder with reading material. Due 6th of October.
Learning outcomes: Modeling complex interactions between an infinite number of degrees of freedom using the harmonic oscillator can give fundamental insights into material properties. An alternative, phenomenological, approach to deducing effective Hamiltonians based on ideas of symmetry can give more general, but less specific, insights into materials in certain asymptotic limits. Calculus of variations and its application to extract dynamical laws from energetic statements.
4. Dimensional analysis (the hardest thing of all) - The famous Taylor example
Learning outcomes: Exposure to the idea that dimensional consideration alone, up to a few situation specific numerical factors, can give suprising insights into the gross/qualitative static and dynamical properties of a systems. We will keep revisiting this fundamental idea/tool throughout the course.
5. Low dimensional systems and tippings points
Learning outcomes: Dynamics in systems with a few coupled degrees of freedom can lead to a zoology of emergent dynamics: Bifurcations, Nonlinear ODEs.
Related movies and possible project ideas: I am legend (the mathematics of SIR modeling for spread of diseases), Alien versus Predator (the mathematics of complex ecologies), 2012 (the mathematics of tipping points in climate modeling)
HW2: Assignment. Excellent set of lecture notes with a chapter on Lagrangian formalism. Lectures from Feynman, almost verbatim, on the principle of least action, some simple lecture notes on the Euler-Lagrange equations, and the wikipedia entry on phonons is very nice!
6. Essentials of stochastic modeling
Learning outcomes: Particulate and more effective ways of modeling stochastic process. Basic probability will be reviewed, Master equation approaches to random walkers, effective Langevin descriptions, and emergent Fokker-Planck formalism. Central limit theorem.
7. Continuum modeling
Learning outcomes: Discussion of assumptions underlying a continuum approach to understanding the emergent behaviours of macorscopic systems/material. Formulation of continuum statements of momentum and mass balance. Discussion of constitutive material properties and the differences between solids and fluids.
8. Macroscopic symmetry breaking: Buckling, convection, and turbulence
Learning outcomes: Application of many of the above ideas to non-trivial dynamics in macroscopic systems.
Some question you might have...
A) What is a mathematical model? (WATCH THIS INTERVIEW) There is a lot of philosophy that can go into answering the first question, but for me, a mathematical model is a map that allows me to logically connect a set of assumptions to a set of outcomes. If done well, mathematics imposes a kind of logic to the deductive process that simply using words does not.
B) Why might a mathematical model be something you want to construct? The reason, I think, why you might want to construct a model is to figure out what is cause and what is effect. Models are almost always trying to make an inference. Step 1: You see some phenomena out there in the world, or in a lab. Step 2: Based on prior work, intuition, and discussions with experimental colleagues, you have a guess as to what the underlying cause is. But usually, its not straightforward to assess whether the mechanism you just guessed would indeed give rise to the phenomena you are observing. So what you do is try to describe your potentially causal mechanism in a mathematical manner. Step 3: Calculate the consequences of your assumed causal mechanism, and Step 4: compare to data. If your predictions and experiment do not match then you are wrong, and you are back at Step 2. That is essentially what a model is and why you might want to construct one. Its because you want to know why something happens. Caution: The wrong assumption could have an effect that looks similar to the phenomena. This is why you typically calculate many independent effects of a single causal assumption/starting point and compare all of them to the observed phenomena. Then the probability of you tricking yourself into thinking you have the right thing, when you really don’t, decreases ... one hopes.
What constitutes an explanation? (WATCH THIS INTERVIEW) This raises an interesting point when considering phenomena that are, in some senses, not fundamental. For example, consider the freezing of water, which is an example of a collective phase transition – note, in case you never have before, that nothing changes about the water molecules when liquid water freezes. It really is something about the packing of these identical particles in space that is manifestly distinct when liquid water freezes. Now there is no doubt that indeed the laws of quantum mechanics underpin the mechanism of freezing/melting. So the problem is solved! In fact all the problems of chemistry/biology are already solved from this point of view. There are indeed people that hold this view point. This brings us to an interesting aspect of theorizing/modeling complex emergent phenomena: Thus the question, what constitutes an explanation? To make it clear, ask yourself why are some material magnetic and some materials not? Why are some insulators, while others are conductors? The answer that its all quantum mechanics doesn’t really help you understand why some material or phenomena has the behavior it does. It just tells me that all materials are made of some similar building blocks and physical laws. It would be like never studying architecture, and just saying its all cement/bricks/stone etc. That really isn’t useful.
C) What kind of model might be appropriate for a given situation? The question of what kinds of models have been used, or might be useful, is a much trickier question than what we have addressed thusfar. There is no “correct” model. There can however be more useful models in a given circumstance. Or perhaps even more agnostic is that different models, alternate mathematical descriptions, for the same phenomena might help address different questions. Now when I mean different models I don’t mean different guesses at causal mechanisms. What I mean is fundamentally different manners in which you can describe the phenomena. For example, a deterministic versus probabilistic approach, an many-bodied approach or a mean-field approach, a continuum approach or a particulate approach etc. etc. A helpful analogy, I hope, is to think about your different senses. For many things out there in the world you can see it, hear it, smell it, touch it, and even taste it. Each one gives you some view/lense into the thing, but no individual one can ever give you the whole pictures. It really depends on what you want to learn about the thing. It’s the same with models. Different kinds of models are like different senses, it’s a different modality through which to view a phenomena.
The phase diagram above is my attempt, surely incomplete and error prone, of trying to describe mathematical models in terms of three features: Degrees of freedom (the number of constitutient things that are playing out some dynamics), the interactions between the degrees of freedom (for example electrostatic forces, hard-sphere repulsion, etc.), and the level of noise/stochasticity in the system. This last one is of course tricky since I have lumped a large number of degrees of freedom into a "noise" source, but lets not worry about this. It turns out this is a useful way of categorizing models.