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Mathematical Moments 160: Exploring Thermodynamics with Billiards

8-ball sitting on a pool table with a cue stick in the background

If you've ever played billiards or pool, you've used your intuition and some mental geometry to plan your shots. Mathematicians have gone a step further, using these games as inspiration for new mathematical problems. Starting from the simple theoretical setup of a single ball bouncing around in an enclosed region, the possibilities are endless. For instance, if the region is shaped like a stadium (a rectangle with semicircles on opposite sides), and several balls start moving with nearly the same velocity and position, their paths in the region soon differ wildly—chaos. Mathematical billiards even have connections to thermodynamics, the branch of physics dealing with heat, temperature, and energy transfer.

Typical models assume that the ball bounces off the walls at the same speed and same angle at which it hits them. But in recent years, mathematicians have added an element of randomness to the mix. Determining the angle of reflection randomly after each collision is a way to model microscopic channels with rough edges. And having “hot” (or “cold”) walls that tend to speed up (or slow down) the ball provides a new perspective on important concepts in thermodynamics. One is the “arrow of time”: Microscopic processes that can be undone lead to human-scale processes that cannot be undone. Although most random billiards models are much simpler than any real-world situation, they are a stepping stone toward advances in tiny technology—even if they don’t help you sink a tricky shot the next time you play pool.

Tim Chumley explains the math of random billiard models and their connections to thermodynamics.

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