chaos – Bean Thinking

Coffee, chaos and computing

Have you ever noticed drops of coffee skipping across the surface of your coffee as you have been preparing a V60? Or watched as globules of tea dance on the resonating surface of a take-away dragged across a table top? The dancing drops can be seen in this video of coffee being prepared in a V60:

These droplets are the result of some fascinating physics. Although we have encountered them on the Daily Grind before (here and here), the more physicists study them, the more surprises they throw up. While the droplets can be considered particles, they are guided around the coffee pot by the surface waves they create as they bounce. In a sense they are a macroscopic example of wave-particle coexistence. There is a significant temptation to explore whether they have relevance for the concept in quantum physics of wave-particle duality. But another aspect of this wave-particle coexistence has recently been shown to produce a different and unexpected connection. A connection between chaos and computing. And as you can create these droplets in coffee, perhaps we could say a connection between coffee, chaos and computing.

floating, bouncing drops — Drops of water can be stable on the water’s surface for much longer than 1 minute if you put the water on a loudspeaker, more info on how to create these at home here.

It is fairly simple to create these surface droplets in coffee at home. The secret to getting stable droplets on the surface is to create a vibration, a wave, on the surface of the coffee liquid. The droplets that then form on (or are introduced to) the surface ‘bounce’ on this wave. If you wanted to create surface droplets reliably at home, you would put your coffee on a loud speaker. I suspect that the reason that they appear in a V60 is that the first drops set up a standing wave on the surface of the coffee that acts to support later drops as they encounter the surface. If anyone has a different theory, please do let me know.

But how is it possible that these bouncing droplets connect chaos theory and computing? It is a consequence of the way that the globules of coffee on the surface interact with the waves that guide them around the coffee. Consider for one moment a particle bouncing around a confined space (the traditional example is of a ball on a billiards table). On an ordinary table, the billiard ball will behave quite predictably, start it off aimed roughly at the side of the table and it will bounce in an easily describable way. But if you make the ends curved or put circular objects in the middle of the table for the ball to bounce off, small differences in initial direction can result in large differences in the final path of the ball (for more details and an animation see here). The billiard ball behaves chaotically, and the initial path cannot be found from the final position, there is no way to re-trace the path of the ball, it is not “time-reversible”.

science in a V60 — A still from the video above showing three drops of coffee on the surface.

The droplet bouncing on the liquid surface appears to move chaotically, just as the billiard ball on a circular table. However, unlike the billiard ball, the droplet is not a mere particle, but a particle linked to a self-generated surface wave. Each time the droplet bounces on the surface, it creates a small wave, like ripples on a pond. The path taken by the droplet is a complex interaction between this self-generated wave, the vibration keeping the droplet bouncing and the droplet itself. This means that if you are able to shift the phase of the bounce by 180º (meaning, that rather than bounce on an upward motion of the surface, the drop bounces on a downward motion or vice versa), the bouncing droplet not only reverses the direction it travels in, it retraces its path. Rather than behave as the chaotic billiard ball, the path taken by the seemingly chaotic globule of coffee can be exactly reversed.

Which is where the link with computing comes in. It is as if each “bounce” of the droplet “writes” information on the surface of the coffee in the form of a wave. The subsequent bounces “read” the information while the reversal of the direction of the bouncing droplet “erases” the stored information by creating a surface wave opposite to the initial one. The authors of the recent paper suggest that “in that sense [the walking droplet can] be termed as a wave Turing machine”, giving the final link to computing.

Whether or not this turns out to be useful for computing is, to me, almost irrelevant. What is interesting is that such a simple phenomenon, that anyone who makes pour-over coffee should have seen fairly often, is linked to such complex, and fundamental physics. If you would like to read more, there is a great summary article here while the actual paper is here.

Perpetual motion in a coffee cup

V60 from Leyas — Could your coffee be used to power a perpetual motion machine?

There can be no such thing as a perpetual motion machine right? Yet less than two hundred years ago it seemed possible that there could be. Not just that, the source of this perpetual motion machine was in your coffee cup. How would you explain Brownian motion?

Brownian motion is the random movement of small bits of dust or coffee/tea particles on the surface of your brew. To see it, you may have to use a microscope though you should take care not to confuse Brownian motion with motion caused by convection currents. There will be Brownian movement even a long time after the coffee has got cold. What causes this continuous movement? When he observed it for the first time in 1827, Robert Brown (1773-1858) had thought it was to do with a ‘life force’. He had been observing pollen suspended in water and noticed that the pollen kept moving under his microscope lens. In 1827, this was a very reasonable explanation, after all, weren’t several people looking for a motion, a force, that gave life?

Sphinx, Brownian motion — Brown used some dust from the Sphinx (shown here with the Great Pyramid) to show that ‘Brownian’ motion could occur in inorganic materials. Postcard image © Trustees of the British Museum

So, he checked if he saw the effect in pollen that was one hundred years old (he did) and then in truly inorganic matter, he looked at the dust from a fragment of the Sphinx. Again he saw the dust fragment move in the water. He had therefore shown that it was not associated with a life force but was something that happened for every small particle suspended in a liquid. What was driving it?

Without knowing what caused it, some people in the nineteenth century had already suggested a device to exploit it, using tiny levers to carry the energy from this continuous motion into devices. Others insisted on finding out what was causing the motion but it was here that the physics of the day hit a philosophical problem. It was proposed that molecules in the water could be hitting the dust on the surface and moving the dust in seemingly random directions. And yet there is a problem with this explanation. At that time there was no way of seeing or measuring molecules. How could physics postulate a theory – or suggest a reality – that could not be tested?

Nasa, Norway, coastline, fratal — How long is a section of coastline? Coastlines can be described as fractal like. Mathematics that grew out of studying random walks and Brownian motion. Image credit NASA Visible Earth/Jeff Schmaltz

An answer came one hundred years ago in a paper published by Albert Einstein (1879-1955) in 1905. In it he made some mathematical predictions that, for the first time, allowed the theory (that it was molecules causing Brownian motion) to be tested by experiment. Jean Perrin (1870-1942) of the Sorbonne, Paris, was the experimentalist who, by careful observation of droplets of water containing a pigment used by water colour artists, provided evidence for Einstein’s theory of Brownian motion. The experiment was so important that Perrin later wrote “.. the molecular kinetic theory of Brownian movement has been verified to such a point in all its consequences that, whatever prepossession may exist against Atomism, it becomes difficult to reject the theory.”

The consequences for our world have been profound. The mathematics that describes Brownian motion is that which we use as the basis to predict the movements of the stock exchange. Extensions of the mathematics have been used to develop new areas of mathematics such as fractals. Even art has grasped the theory of Brownian motion, the Anthony Gormley sculpture “Quantum Cloud” is based on mathematics describing Brownian motion. Everywhere you look there are phenomena described by the movements in your coffee cup. What we have yet to do is find that perpetual motion machine.