James Gleick, a pretty prolific technical science historian and author, wrote a book called Chaos in 1987. Coincidentally I found this book whilst browsing the basement of the Cinema bookstore in Hay-on-Wye a few months ago.
One of the chapters mentioned the Lorenz water wheel, an example of a system that is in-principle deterministic, but exhibits non-periodic oscillations that appear random. Another property of this system is that if you traced out its phase space, by plotting the value of observable quantities in real time, you'd find that the line you trace out never overlaps on itself. Harvard did a real life build of this experiment here.
This is a simulation of one made by Claude
I find such demonstrations of chaotic phenomena really interesting because they seem to happen in this weird area of physics that produces random-like behaviour from deterministic laws. People in modern times might like to call it emergent complexity, other people call it chaos, but I personally think that each different kind of system that exhibits behaviour like this is unique and cool, so theres no point giving a single label! Other things that make complex behaviour from simple rules are cellular automata like Conway's game of life.
To be clear, there is overwhelming evidence that nature, in some aspects, is genuinely random, in the deepest sense of that word. Not like the complexity that causes a dice roll to appear random; I mean genuinely indeterminate. To the best of our knowledge, quantum mechanics is fully indeterminate - if you prepare an electron in a neutral state, then you would observe it either spin up or spin down with equal probability. But just because the micro-scale atoms indeterminate through quantum mechanics, it doesn't mean that we can't predict things. Thermodynamics allows us to make general rules from aggregates of many atoms, like the ideal gas law.
Chaos, on the other hand, works in the opposite direction. Classical physical laws like Newton's theory of gravitation and relativity are determinate in their formulation. But from them, we still can get chaotic phenomena - the Lorenz water wheel, the three body problem, and much more, which behave essentially at random even though they obey clockwork physical laws. And so, because we so far have not been able to accurately forecast chaotic things, there is subjective uncertainty that humans have to deal with when it comes to forecasting chaotic systems. This subjective uncertainty is distinct from the genuine indeterminacy that we think govern quantum systems (although admittedly, its hard to distinguish the two).
This mix of the subjective, observer dependent view of uncertainty alongside objective physics is something I want to think about a bit more. I've been reading more about Bayesian probability and how subjective probabilities are involved in uncertainty of measurement. I also learned recently from David Spiegelhalter's book *The Art of Uncertainty* that institutes in metrology (the science of measurement) also allow for the inclusion of something called Type B uncertainty, an uncertainty that comes from judgement rather than from repeated measurements. I would like to get a handle about distributions of observables in chaotic systems. Indeed, Lorenz himself was a meteorologist!
I am trying to find demonstrations of such complex phenomena on a table top. I've written a list of cool experiments which might fit this kind of bill in a previous post. In the best case, maybe I might be able to discover some dynamics that are overlooked. Failing that, it's still cool to try and find stuff that can be used as a demonstration. For example, the traditional chaos demonstration with rotating buckets of water feels quite impractical.
In my last post I got claude to make a list of similar kinds of experiments using the list of equipment that I assembled in my lab. One of the things that caught my eye was observing convection currents and the plumes that you get when you apply a temperature gradient to a fluid. Previously I visually looked at Rayleigh–Bénard convection cells with mica powder and silicone oil, but I wanted a more quantitative way to track convection phenomena, so I tried this experiment.
Conduction is when heat is carried across stationary matter, like a metal. Convection is when heat is carried by parcels of moving matter, which happens in fluids. When the heat gets high enough, things become turbulent and unpredictable. But I wanted to straddle an area that wasn't too crazy, with some dynamics that are regular but not too regular. So I wanted to see if I could measure what I think are convection plumes that would come with slowly heating up a beaker of water. You apply heat at the bottom, and then you have parcels of water that are carried in these plumes which rise due to the decreased density of water, which causes the temperature measured to fluctuate.
I think this is cool because we always assume things heat up evenly when you apply heat to something. But that's not true, there seems to be a whole set of indeterminate phenomena that you can get in as simple as heating a cup of water! Biosecurity / building engineers / weather people / chefs take note. I also think there is this plume-type behaviour that you get in the Earth's crust, but I don't know too much about that.
An infrared camera is too expensive for me. So the experiment I decided to test if plumes occur was a lot cheaper and a lot simpler. I first got a pair of thermocouples, and then wired them up to a set of adafruit thermocouple chips. Thermocouples convert temperature into a differing electric current, and are one way of probing temperature. And then I decided to put one above the other in a heated beaker and then try to record the temperature difference. If there was a plume, we would see a temperature dip as the plume dissipates. Here is a simulated animation of the convection plumes that I wanted to capture;
And here is what I actually ended up building
Here are the results when I ran the experiment. The first graph shows the temperature of the thermocouples separately, and the second graph shows their difference. The blue line is the temperature of the bottom thermocouple nearest to the heater, and the red line is the one at the top.
In the diagram, you can see the oscillation in the temperature difference. It wangs around, even becoming negative, showing that the tempeature at the top at some points becomes higher than at the bottom.. This is evidence for what I think is convective behaviour, because otherwise, we should see the temperature difference being mostly constant. Whilst the amplitude of the oscillations look quite janky, the period looks kind of regular, which I think is cool, but there is no real pattern (note to self to check on this later - maybe there are cool patterns to be found here).
There are some subjective uncertainties here though. Are these spikes the result of some noise or systematic error? Seems unlikely, because the noise in the initial measurements when the heating was off is small, but who knows. Also, something that could be causing the oscillations could be the heating duty cycle of the heater itself - after all, it's a cheap heater that I got from China. So next time I'm going to try and check this with open flames under a candle.