# On the laws of universality or what Tetris can explain to us about coffee stains

Original author: Aatish Bhatia
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The next morning, after the big snowstorms that swept the northeastern United States, I was sitting in my car, ready to challenge the dangerous road conditions, to go to a local cafe. My home in New Jersey was outside the main storm path, so instead of snowdrifts we were greeted with a mixture of wet snow and freezing rain. And sitting in my car, I could not help but be fascinated by these strange patterns of ice formed on the windshield. Here is what I saw:

When I watched this miniature world, created on the windshield as an alien landscape, I thought about the physics of these patterns. Later, I learned that these patterns of ice are associated with a very active field of research in mathematics and physics known as universality . The basic mathematical principles behind these intricate patterns apply to some unexpected things, such as coffee rings, the growth pattern of bacterial colonies, and the movement of flame on cigarette paper.

Let's start with a simple example. Imagine a game similar to Tetris, but with only one type of block - 1 x 1 sq. These identical blocks fall at random, like raindrops. Question: which block pattern do you expect to see as a result?

You could assume that since the blocks fall randomly, everything will end with a smooth, uniform pile of blocks, like piles of sand that gather on the beach. But this does not happen. Instead, we get jagged horizons where tall towers stand next to deep basins. Creating a high stack of blocks next to a low stack is just as likely as creating a next to another tall stack.

This is not very similar to what I saw on the windshield.

This tetris-like world is an example of what is known as the Poisson process, which I wrote about earlier. The point is that randomness does not mean uniformity. Instead, the mess is clusters like the jagged horizon of Tetris blocks that you see above, or likedistribution of Humming bombs that fell on London during World War II .

This Tetris example may seem a bit abstract, so let me introduce you to a guy who takes abstract ideas and connects them with real-life examples. His name is Peter Junker , and he is a Harvard physicist.

Junker was curious about the causes of the ring stains of coffee. In 1997, a group of physicists found outwhy the coffee stain forms a ring. As the coffee evaporates, the liquid from the center rushes to the edge of the drop, capturing coffee particles along with it. The drop begins to level out. In the end, a thin ring remains, as the coffee particles all rushed to the edge of the drop. Here is an amazing video shot by the Junker team, which shows how this process looks.

What Junker demonstrated is really very elegant. He found that the reason that the coffee drop forms a ring is because of the shape of the coffee particle. Look at a drop of coffee under the microscope and you will find small, round caffeines. If you look closer at the edge of the coffee drop, you will see particles gliding past each other, like blocks in our Tetris . In fact - Junker showed mathematically that the nature of the accumulations of these coffee particles is exactly the same as our randomly falling Tetris blocks!

But here is what is amazing! Junker and his colleagues also found that if we replace the spherical particles with more elongated ones like ovals, we get a completely different picture. Instead of a ring, we get a solid spot. The video above shows how this happens.

In one case, we get a ring, and in the other, a solid spot. So why does the shape of the particle change the overall picture of growth? To understand why oval particles behave differently, we will customize our Tetris game. Let's name the new version - Sticky Tetris.

In Sticky Tetris, the block continues to fall until it touches another block that stands in place of at least one side. As soon as the falling block comes in contact with the non-falling block, the first one immediately freezes in place.

This is a small change in the rules, but it has rather big consequences. In ordinary Tetris, you need many blocks to fill deep gaps, in Sticky Tetris, you can fill a large gap with one block. Very quickly, the difference in height between the towers begin to level out. Instead of the rough, jagged horizon in ordinary Tetris, the horizon in a sticky version is a smoother world.

It looks a lot more like a windshield pattern!

And here is the point. While spherical particles behave like ordinary Tetris blocks, oval particles behave just like Sticky Tetris blocks. At that moment, when the moving oval coffee particles touch the motionless, they stick in place. Instead of jagged horizons, we get this Swiss cheese, a complex structure of sprawling threads, separated by holes and crevices.

Thus, we have two significantly different types of growth processes. On the one hand, we have particles that accumulate like ordinary Tetris blocks or like caffeines in a coffee ring. Here are animations of real data from Junker's lab showing how it looks.

On the other hand, we have particles that accumulate as sticky blocks or like oval caffeines. The growth of these particles is as follows (again, this is real data).

Obviously, these are two qualitatively different patterns.

But they alsoquantitatively different. Remember that in the world of ordinary Tetris, we end up with a jagged horizon, while in the world of Sticky Tetris, the horizon is smoother. By studying how the top layer of particles (horizon) expands over time, physicists can classify growth processes. In field jargon, processes that grow at different speeds belong to different classes of universality .

Let's say you study how ice particles thicken on your windshield. If the speed with which the horizon expands corresponds to the blue curve in the graph above, then the ice thickening process is in the same universality class as Tetris. If it coincides with the violet curve, then it falls into the sticky Tetris universality class.The key point is that many seemingly different physical systems, in mathematical analysis, show identical growth models. These slightly mysterious tendencies for the same behavior of very different things reflect the essence of universality .

Moreover, there is a rich mathematical theory behind the sticky Tetris universality class described by an equation known as the Kardar-Parisi-Zhang equation (KPZh).

This deep connection between coffee rings and the CSW equation took Peter Juncker by surprise. According to Juncker: “ Alexey Borodin, a mathematician from MIT, contacted us after we published an article on how the shape of particles affects their deposition in a coffee ring. He watched our experimental videos online and recalled the modeling he was doing earlier. We would never have begun to study this area if Alexey had not attracted our attention to it. ”

And this sticky Tetris universality class arises in all sorts of strange places. An example is burning paper. During a 1997 physical experiment , carbon paper was set on fire at one end and the flame front was fixed while it burned. Here is a sketch of what happened.

The flame develops smoothly, with a wavy pattern as it burns through the paper. And when physicists studied in detail the behavior of this flame front, they found that it exactly coincided with the predictions of the CSW equation. They repeated their experiment using tissue paper and obtained the same results.

And another very unexpected and elegant example is bacterial colonies. A group of Japanese physicists in 1997 showed that in a certain nutrient medium the edge of a bacterial colony grows exactly in the manner predicted by the universal class of CSW. Here's an animation of it in action. You are looking at enlarged photographs of the edge of a bacterial colony as it grows in a Petri dish .

Nowif you think about it, you will find something deeply mysterious . Colonies of bacteria, traveling flames and coffee particles are all completely different systems, and there is no reason to expect that they must obey the same mathematical laws of growth. So what is behind this mysterious versatility? Why do such different things play by the same rules?

You may have noticed that all these examples look a bit fractal . It turns out that the phenomenon of universality is inextricably linked with the fact that these systems are self-similarlike fractals. As I scaled down to capture ice particles on the windshield, the overall picture looked the same. The same is true for the flame front, the edge of a bacterial colony or the sticky Tetris horizon. Here is an example of a self-similar curve (orscale-invariant , as physicists like to call it).

Surprisingly, this self-similarity means that many of the smallest details, such as bacteria or caffeines, do not change the overall picture. According to Peter: “The fractal nature of these growth processes is important for their universality. In order to be universal, the system cannot depend on microscopic details, such as particle size or scale of interactions (this does not apply to form - approx. Transl.). Thus, a universal system must be scale-invariant. "

Which brings me back to the icy particles on the windshield. They stick together in these wonderful fractal patterns, which, in my opinion, are very similar to Sticky Tetris. I wanted to know if there is a connection between these ice particles and the KPZ universality classes, and I asked Peter Juncker a question.

He replied: “These videos are fantastic. I agree with you that the main process taking place here is very similar to the CSW process. This can be a great example of why it is difficult to determine CSL processes in real experiments. Changes in these structures have a strong impact on the overall development of the system. But it is likely that this system has the same growth rates as CSF processes. ”

The part of physics that makes these ice floes short-lived makes them difficult to study. And so, let me end with a short video on growth and longevity;)