"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt." -- Bertrand Russell

Monday, August 1, 2011

Complexity: The Human Eye and the Ice Cube

The human eye is a rather marvellous object. Somehow, billions of atoms know how to come together, to attach to each other, folding up into proteins, those proteins somehow know how to form cellular structures, and those cellular structures somehow know how to form this complicated optical apparatus that we call an eye. Now, I am certainly not a creationist, though I understand a little bit about where they are coming from. How is it that something which is so mind numbingly complicated can arise from entirely natural processes? How do the atoms know how to form into proteins and other complicated organic molecules which, in turn, know how to form all of the myriad parts of things like eyes or noses or even hair? It seems almost as if there must have been some kind of supernatural power -- a God if you will -- that intelligently designed it.

Truthfully, it seems absolutely miraculous. But really it's not; the complexity of nature is not an illusion, but that this complexity is miraculous is illusory. Somehow, nature is able to produce complex structures by the repeated use of very simple rules. In this post, I'd like to explore a much simpler complex system, but I hope that this analogy will help to shed some light on how nature is capable of producing very complicated things by the repeated use of very simple processes.

What I'd like to talk about are phase transitions. The first question that one might ask is "what is a phase transition?" Probably the three most ubiquitous examples of phase transitions are boiling, melting, and freezing. There are other examples, and my PhD work is on an abstract mathematical system which has phase transitions in it, but I would like to stick with what people are most familiar with.

Therefore, let's talk about freezing. It seems like a relatively boring subject; if you stick water into the freezer, it turns into ice. I would not be surprised if you responded by asking me "what's the big deal?"

But let's think about this for a moment.

If you lower the temperature of liquid water, there is a sudden point at which the water collectively changes form. Somehow, every atom in the water knows to form ice. You might naively think that the water begins to be more and more ice-like as you lower the temperature, and eventually it just becomes ice. But you'd be wrong.

At that temperature, it just becomes ice. Lower than that temperature, it simply is ice -- and higher than that temperature it's water. Actually, this story is a little bit too simplistic and I'll explain why momentarily. But before we get to that, just think about this.

Somehow, at that temperature, the water collectively changes form. Ice has a very different atomic arrangement from liquid water. In liquid water, the atoms are free to move around. Unlike a gas, they do interact with each other, but only locally. And because they interact only locally, there is no large scale order in the system.

Think of a crowd of people moving on a city street. Each individual person might interact with a person next to them. But the entire crowd does not produce a collectively ordered arrangement. Ice, on the other hand, is something like people sitting in a crowded stadium. Each person is located in a seat, and the arrangement of the seats collectively orders the people.

Of course, in liquid water, there are no seats. Instead, the atoms interact with each other, and they do so in such a way that the energy of the system is minimized. Why does it minimize the energy? It does so because any arrangement of the system in which the energy is higher would be unstable. It would spontaneously relax into a configuration in which the energy were minimized.

One way to think about this is the following. Remember that when you were a kid, you probably tried to dig a hole in the sand at the beach. You might also remember that no matter how big of a hole you dug, nature always won. The hole would collapse and fill with water. Erosion would destroy the side walls. You can ask why nature prefers not to have these holes, and, in the case of the beach, the answer is that nature prefers the smooth flat surface of the beach to any holes that any pesky kids might try digging. Things in nature have a tendency to fall apart, until they can fall apart no farther. These are the configurations that minimize the energy, and in the case of ice, the configuration which minimizes the energy is an ordered atomic arrangement, like people at a movie theater, order prevailing. For anyone who is interested, the particular kind of energy that is being minimized here is what we call the free energy, and the tendency for nature to sort of fall apart until she can do so no longer is what we call the second law of thermodynamics. It is closely related to what we call the entropy, which is the measure of nature's disorder.

This actually brings us to our next point. Remember that I said it was actually too simple to just say that the transition occurred at a definite temperature. Actually, when you go past the freezing point, you still have liquid water. However, that liquid water is unstable. Any slight disturbance, any slight jostling, will be enough to kick the system into ice. You might have seen youtube videos where people put a bottle of water into the freezer with the cap on, then removed it from the freezer, they shook the bottle, and the water froze spontaneously. What happened was that the liquid water was highly unstable. The slight disturbance of removing the water from the freezer was just enough to freeze it. This is called super-cooling, and the same thing is possible at the boiling point -- in which case we have super-heating. If you put distilled water into the microwave, and then remove it and disturb the water, the disturbance is just enough to force the water to boil. Unfortunately, it will do so explosively and often this results in horrible burns.

I'd like to explore one last thing. Remember that when the water freezes the atoms form that highly ordered arrangement, just like people filling in a theater. However, unlike the theater, the atoms do not sit in rectangular rows. Rather, because of the forces between the atoms, they form a hexagonal pattern. And that hexagonal pattern results in the beautiful arrangement that you see in individual ice crystals, such as snowflakes. What you might not know is that water always freezes into those crystals. With an ice cube, however, the crystals have all been fused together so that with the naked eye you can no longer tell them apart. However, if you had an eye that could see deep down within the structure of an ice cube, you'd find that it was a highly complex network of hexagonal structures. In fact, most solids are constructed from crystal structures of some kind. Any piece of steel that you've ever picked up is similarly assembled.

At any rate, the main point that I would like to leave you with is that nature is capable of producing highly complex structures by simply iterating simple rules. I haven't, of course, answered the question that we started out with; that is, I have not explained to you how biological complexity is produced. Nonetheless, I hope that you can appreciate that biological complexity is simply a different manifestation of the same idea. And I hope that you will never look at an ice cube in the same way ever again.

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