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Mathematics and Probability 

Start with two containers, labeled A and B, and a bucket load of chips numbered uniquely from one to one hundred. In container A, place one hundred balls, each labeled uniquely with numbers from one to one hundred. Container B starts out empty. Pick a chip at random from the bucket and read its number N. Move the ball numbered N from the container it is in to the other container. Replace the chip and repeat the process, each time picking a random chip.

What will happen over a long run of time?

Author: Joseph Mazur

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Poincaré's general theorem predicts what will happen. The number of balls in container A will decrease at an exponential rate until both containers have approximately the same number of balls. As the number of balls in container A decreases, the likelihood of picking a chip marked with a number from A decreases. The rate of decrease is proportional to the number of balls remaining in container A; that's why the rate is exponential. But, with absolute certainty, all the balls will eventually return to container A, although it might take an enormously long time for that to happen. The second law of thermodynamics tells us that you can play the same game with gasses. Take two containers, one with gas at some pressure and the other empty. Connect the two by a tube that lets the gas move freely between them. The gas will quickly spread until both containers have half the starting pressure. This equalization of pressure is one example of a universal tendency of particles to distribute themselves in as many ways as possible. We measure this tendency with a variable called entropy.  The gas molecules will randomly bounce off each other like bubbles in a pot of boiling water so that, over time, each will find itself, for a while, back in the container it started from. Henri Poincaré demonstrated this in a general theorem about dynamical systems.