Sunday, November 22, 2009

Rules or Consequences

Can we learn about one phenomenon by studying a completely different one?

Putative "laboratory versions" of exotic phenomena appear regularly in the news, such as microwave analogs of "rogue" ocean waves, optical-fiber analogs of rogue waves and black holes, and, as I've discussed here, magnetic-crystal analogs of magnetic monopoles.

But not all of these experiments are equally illuminating. Researchers, and journalists who write about them, need to think clearly about how the two systems are related, and what's missing. Experiments on a model system can show what behavior arises from shared underlying rules, and how that behavior changes as conditions change. But only experiments on the original system can test whether those rules are relevant.

The results of known mathematical rules aren't always obvious. Even Newton's second law, which relates the force on an object to its acceleration, only stipulates a differential equation that researchers must solve to find how an objects position changes with time. When the force is constant, this is easy: the position follows a parabolic course in time.

For more complicated situations, scientists often can't relate the rules to the end result. In some cases they turn to simulations, which can be regarded as a model system that, ideally, embodies the mathematical rules perfectly. But simulations are often restricted to unrealistically small systems that could behave differently than the real McCoy.

In these cases, researchers can learn from actual systems that--they think--follow similar rules. For one thing, this may make precision measurements easier. Placing a block on an inclined plane, for example, slows down its acceleration due to gravity, making it possible to test the parabolic law more precisely.

Unfortunately, the model system may introduce complications of its own. The friction on a sliding block is significantly different than that air friction on a falling body--for example it's much larger before the block starts to move. Even though the rules of gravitational force are the same, the differences may completely obscure the relationship between the two systems. Researchers must then spend a lot of energy tracking down these differences.

But to draw any parallel between two systems, researchers must establish that both are governed by similar rules. Unless they know that, seeing a particular behavior in a model system, by itself, is irrelevant for deciding if the original system follows the same rules. The way to test that--but not prove it--is to do experiments on that system, and see if the behavior is similar.

In our example, if an object follows a parabolic time course, it might well be that it is responding to a constant force. (Of course, it may just be moving through curved spacetime.) With luck, the model system--the inclined plane--would have demonstrated something close to this parabolic result, even if the equations had been unsolvable. The model system then hints at a similarity of the governing rules.

Similarly, a chaotic microwave cavity or an optical fiber might show a "long tail" in the distribution of wave heights that mathematically resembles that which is experimentally measured on the ocean, and which occasionally spawns mammoth rogue waves. Because it's easier to vary the conditions in the laboratory, these experiments might also show what aspects of wave propagation are relevant to rogue-wave formation. In these systems, researchers already understand the basic features of wave propagation--the question is what happens when they combine the ingredients in various ways.

In contrast, physicists do not know whether the basic equations of physics allow magnetic monopoles. Some grand unified theories predict them, but they've never been seen in free space, despite extensive experiments. The observation of monopole excitations at low temperatures in magnetic materials called spin ices has absolutely no implications for the nature of the fundamental equations. It may be that it helps to understand how "real" monopoles would behave, if they exist. But it says nothing about whether they do.

Model systems can reveal important relationships between models and behavior. They can also uncover real-world complications that need to be included to make models more relevant. But to find out whether a model applies to a particular system in the first place, researchers need experiments on that system. Experiments on a model system aren't enough.



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