Wednesday, October 4, 2006

The chameleon in a mirrored box

The biologist Stewart Brand posed this riddle to a fellow biologist in 1970s: What color would a chameleon take when placed on a mirror? The riddle is in a certain sense more interesting as a metaphor than as an actual biological question about that lizard.

The actual lizard changes its color depending on how it interprets the context. "A chameleon changes color out of a fright response so it all depends on its emotional state," pointed out Peter Warshall. "It might be frightened by its image at first, but then later 'warm up' to it, and so change colors."

Kevin Kelly actually did the experiment with an anole, a cheaper type of lizard that can also change its colors (although its ability isn't as wide ranging as that of a chameleon), placing it in a box with mirror walls – so it wouldn't just walk off the mirror. The lizard always turned green and it showed visible sings of discomfort (when it was taken out of the box and it relaxed it changed into brown). Supposedly, it has chosen green because leaves have that color and, not knowing what to do, it changed into the safest thing. When kept in the box it sometimes turned temporarily to a sort of brown but it didn't seem to be capable to relax in there! But how could anyone in its situation?

The theoretical chameleon, on the other hand, is supposed to change its color matching that of the environment. So what would such a chameleon do? Suppose one would design a robot chameleon that would work like the theoretical creature, what would it do?

When Kelly asked various computer scientists to give their best shot, they either speculated that the chameleon would randomly change its colors never settling into one or that it would gradually reach some color – although each time the experiment is done the end color would be different.

The two imagined solutions reflect two possible outcomes often seen in complex phenomena, and the theoretical chameleon in a mirrored box is a nice model for such phenomena. The two solutions are also the two sides in a famous 19th century controversy surrounding the issue of statistical mechanics.

One idea would be that, as the back of the chameleon imitates the color of the environment, and the mirror reflects the color of the chameleon's belly, there would be a tendency for the back of the chameleon to get the same color as its belly. This would not happen when you place the chameleon on a normal surface – the belly could then take any color whatsoever.

So, in the mirrored box there is a tendency for the color of the entire chameleon to become more uniform (or at least for the back to take the color of the belly). There is a certain "force" in that direction, although this is not part of the definition of how the theoretical chameleon works. This tendency is just a side-effect of how the chameleon works and of the nature of its environment (the mirrored box).

This can be a metaphor for how the second law of thermodynamics works in the real world: if various things, some hotter some colder, are placed in a closed environment (like the mirrored box or, more appropriately, a thermos) they eventually all reach a uniform temperature, although there is no "law of nature" to assure that this should happen – but, nonetheless, that's the observed side-effect.

In the 19th century, Boltzmann has argued that the second law of thermodynamics (the fact that things tend to equalize their temperatures) can be seen as a side-effect of the laws of mechanics. Loschmidt, on the other hand, was the one who argued that this cannot be so, because mechanics alone would only lead to a perpetual kaleidoscope, the temperatures would flicker constantly and never reach equilibrium – so, thermodynamics had to be considered as something additional to mechanics, he thought.

Suppose the chameleon doesn't move, it just stands on its feet, and that its entire belly is red. Then its back would turn red. But suppose half of its belly is red and the other half is green. Then half of the back will turn red and the other half red. If the belly is like a chess board, the back would become like a chess board. The point is that in each case there would be a final, stable, state that depends on what the initial state of the belly has been.

What's a little bit more interesting is what happens to the sides of the chameleon. Here, there is a two-way connection: the left side influences the color on the right side and vice-versa. If a fly rests on the left side, an image of the fly would form on the right side, but that would cause an image of the fly to form on the left side, right under the real fly, thus scaring the be Jesus out of that fly. The chameleon would act like a mirror for anything else in the box, although it isn't really made of mirror-stuff (that reflects light).

Suppose now that the mirror walls aren't perfect, that they diffuse the light a bit. Then, for example a very fine grained red-and-green chess board on the belly might translate into a brown back. The imperfections in the mirrors disturb the information about the color of the belly pixels. But such a chess board on the left side of the chameleon would make the right side brown, which in turn would make the left side brown as well, thus leaving no trace of the original fine grained chess board.

This was Boltzmann's genius idea – that some structures, such as the brown color, are more stable than others, such as the fine chess board structure, and that thus there is a tendency to go from the less stable to the more stable. That's why temperature becomes uniform – because the information about the exact state of each thing gradually gets lost until they all end up in their most stable states, the ones that "incorporate" the least information about the past (have the highest "entropy"). And the stability of each depends on how the others look. For example three ice cubes in a thermos remain three ice cubes (or maybe eventually get stuck to each other forming a single larger ice cube), but one ice cube and a bit of hot coffee in a thermos get transformed into a bit of warm coffee – the end state of each thing depends on the initial states of each of its companions.

Suppose now that the chameleon moves. A key issue here, as Holland remarked, is the "lag time", how fast the chameleon manages to take the information about the environment's color and to put it into use. If the chameleon moves faster than its lag time, its various parts won't manage to get synchronized anymore. The whole chameleon would start to "flicker".

This is analogous to what happens in nature when there isn't sufficient time for a system to reach its equilibrium point, because the external influences are changing too fast. The equivalent to the chameleon's lag time is the speed with which interactions spread throughout the system – it is no wonder that such complex "flickerings" happen easier in liquids than in gases or solids. A great example of such phenomena is the turbulent flow – like the water flowing under a bridge and forming that concert of eddies. The water is flowing so fast that the equilibrium seen in calm water cannot form – there isn't enough time for the uniformization to occur. The transition to turbulence happens only when the speed becomes sufficiently large, it is similar to the chameleon moving faster and faster until it goes beyond its lag time – and then flickering starts.
It is interesting to note that such complex phenomena can appear even if the chameleon doesn't move – even in the case when the environment doesn't change faster than the "lag time" but when the environment just is in a certain way. For example, consider a mirrored box that isn't square, but round (or contains round parts). Such a "non-linear" box has the property of being "chaotic", meaning that two rays of light starting very close to one another can end up after a few reflections very far apart – this could not happen in the case of a square "linear" box.

Such a box might also cause the chameleon to flicker. In the same way as two rays starting very close to each other can end up very far apart, it is also possible for two rays starting very far apart to end up very close. Therefore, the same part of the chameleon is influenced by different parts. But the influences don't all reach their destination simultaneously. If the lag time is very, very small, the chameleon can start to feel these differences and consequently will flicker.

As you can see, while in the previous example the flickering was caused by the very long lag time, here it is caused by the very short lag time! Such phenomena have also been observed, the most famous example being a chemical reaction called the Belousov-Zhabotinsky reaction. This reaction oscillates through various states, which are visible – as you can see in the three images below.

The reaction basically involves A turning into B which turns into C. But what makes it special is that you set the experiment in such a way that you can control the amount of A and of C. This influences the state B. One would expect that there would be a certain equilibrium value for B. But one can set A and C in such a way that B starts oscillating – it never settles in one state. When Belousov first discovered an example of such a reaction in 1950 he couldn't publish his results because people (wrongfully) believed the reaction contradicted the second law of thermodynamics and thus, they reasoned it had to be something wrong with it. Zhabotinsky studied it ten years later and played with various chemicals, but it remained a curiosity until chemists and physicists started to be interested in what happens in systems far from equilibrium. In the case of the Belousov-Zhabotinsky reaction the external environment (the amounts of A and C) is fixed, but nonetheless the system B flickers and cannot reach a final stable equilibrium state.

We can note one final twist to the chameleon story. How do we know what colors the chameleon has? Well, you have to look into the box. But because the box has mirror walls you cannot do it without appearing in the mirror yourself. But this will influence the chameleon! Your face or a portion of it will get imprinted on the chameleon. This image will become involved in all the processes I have just mentioned above. You cannot look at the chameleon in the mirrored box without changing it.

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