Friday, October 07, 2005

On the possibility of prediction

It is often said that predictions are not useful because reality is too complex to predict. Here’s an example of a simple system with completely unpredictable behavior, but a completely predictable fate.

It is obviously not possible to predict the detailed behavior of complex systems. Even simple mechanical systems can have completely unpredictable behavior. One such system is a popular executive desk toy. The toy consists of one magnet that is free to swing on a dangling string or rod, and several additional magnets fixed motionless below it. The fixed magnets are arranged to repel the lower end of the pendulum magnet, which is impelled by the force of gravity to return to the center of the array of fixed magnets. When the executive pulls the pendulum to one side and releases it, a marvelously complicated and unpredictable series of motions ensues. What is not obvious except to those who already know why, is that the motions of this toy are unpredictable in principle, even though we understand completely the rules that govern those motions, and how to compute the trajectory of the pendulum. No matter how precisely we measure the shape and strength of the magnetic fields, no matter how precisely we measure the initial position, direction, and speed of the pendulum, any computer program that computes the trajectory of the swinging magnet according to those initial measurements will predict a trajectory that diverges wildly from the actual trajectory within the first second or two of the several minutes it will take the motion to die down. The motion of the system is said to be “sensitive to initial conditions”. A system that is sensitive to initial conditions hugely magnifies the effects of arbitrarily small errors in the physical measurements of the initial conditions. Two sets of initial conditions that are not exactly the same, but which differ only by arbitrarily small amounts, produce radically different trajectories within a second or two of movement.

To drive the point home, consider that by some cosmic accident, we obtain the initial position, speed, and direction of the pendulum with perfect accuracy, and that the word length of our computer is much longer than sufficient to specify these measurements perfectly. Even in this case, the computed trajectory will differ wildly from the computed one within seconds, because the intermediate positions calculated by the computer will not be capable of perfect representation within the computer – the computer will have to round its computations, introducing tiny errors. The computer will also have to round its representation of the magnetic field at intermediate positions of the pendulum. Because of sensitivity to initial conditions, even the smallest rounding error will be hugely magnified, with consequent failure to approximate the trajectory.
Now suppose I start the pendulum swinging in a box that I hold several feet above the floor. I close the box immediately after starting the pendulum. A couple of seconds after starting the pendulum, I cannot know its position and speed, even with the aid of the most powerful computer. But if I let go of the box, I can be perfectly confident that the box will be on the floor in a second or so, and the pendulum with it.

Conclusion: macrostates of "unpredictable" systems may be easy to predict.

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