I loved the film π. I consider it a hugely flawed film, but what I loved about it was the way that it worked in subtle allusions to the underlying concepts motivating the film. The main character walked through a park and they point the camera skyward to show the denude winter branches of the trees, an example of fractal symmetry. One of the images that they showed a number of times throughout the film was that of a cup of coffee. Whenever someone ended up in a diner, we got a tight-in shot of them dumping the cream into their coffee and the blooms of turbulent fluid redounding from the depths. It’s a perfect example of turbulence, a phenomenon that utterly defies computation. Since π I’ve never looked at a cup of coffee the same. Every time I pour cream into my coffee it’s a little ritual where for just a second I consider the boundlessness complexity of the world, as close as the cup in my hand.
I was amused to see a recent article in New Scientist invoke the image of the cup of coffee in reference to the problem of turbulent fluids in supernovae (Clark, Stuart, “How to Make Yourself a Star,” vol. 200, no. 2679, 25 October 2008, pp. 38-41):
As the dense inner material is flung through the less dense outer layers of a star, it creates turbulence and mixes everything up. Traditional computer simulations do not model turbulence well.
“Our theoretical understanding of turbulence is incomplete,” says astrophysicist Alexei Khokhlov of the University of Chicago. In other words, you cannot write down a set of equations describing the state of a turbulent system at any given time and then use them to predict what it will look like next. Instead, you have to employ a brute-force approach, using sheer computer muscle.
To seen the scale of this problem, take your morning cup of coffee and stir in some milk. You are using turbulence to mix the two fluids. To determine how they mix, physicists mentally split the cup into boxes and assign numbers to represent the properties inside each box, such as the temperature and density of the fluid. A computer can then calculate how each box interacts with its neighbors during one brief instant of time and then re-evaluate those numbers. Once it has done this for every box, it starts again for the next slice of time and so on.
To do this massive computation perfectly, each box should be tiny and contain just one fluid particle, but before you can get anywhere near this sort of precision, the numbers become mind-bogglingly large. Scientists talk of degrees of freedom as a measure of both the numbers of particles in a system and the number of ways each particle can interact with those around it. A single cup of coffee possesses a staggering 1040 degrees of freedom — far more than you can model on today’s computers. “Maybe in 10 years we will be able to fully model a cup of coffee,” says Khokhlov.
Until then the computation will always be approximate, and thus prone to errors, because small-scale physical interactions are not being taken into account. … If it is going to take 10 years to fully model a cup of coffee, how long until we can model an entire star?
“Never,” Khokhlov says. “Not until someone comes up with a cleaver theory that does not depend on what is happening on the small scale.” The only hope is to continue to investigate turbulence to learn how to better approximate its behavior.