A billiard ball in a rectangle with two circular ends: slight changes in initial conditions make such a big difference that eventually it could be anywhere. To be precise, we say its motion is "ergodic".— John Carlos Baez (@johncarlosbaez) July 28, 2018
This animation was made by @roice713.
(to be continued) pic.twitter.com/rQPS0OOdXH
For the precise definition of "ergodic" read my diary:https://t.co/kcgMZNahqv— John Carlos Baez (@johncarlosbaez) July 28, 2018
A billiard ball moving in a convex region with smooth boundary can't be ergodic. In fact, in 1973 Lazutkin showed this for a convex table whose boundary has 553 continuous derivatives!
I appreciate the tag and am glad I get to follow this thread now, but also need to correct the attribution - the animation was by Phillipe Roux :) https://t.co/8HVbdDZ2dW— Roice (@roice713) July 28, 2018
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No particular region of the space is priviliged. No matter where the billiard ball starts, in time it could be anywhere. A bit more exactly (Wikipedia):
In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space. In physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics.