Chaos in a very simple system: Stadium Billiard
Basics:
A point mass (or "light ray") ist started at some point along the
periphery of a "corral", or stadium, comprised of two semicircular
walls joined by short horizontal pieces. All wall collisions are
elastic, i. e. energy is conserved.
Phase space, as depicted in the blue field, consists only of
vx,vy,
and energy conservation confines the state points to a circle.
Ergodicity/Mixing:
Starting a large number of scatterers from the same position but with
slightly varying directions
(variation = one in 1012), we
observe the following:
- for the first few bounces the phase points are indistinguishable
- then the points spread out very fast to cover the entire circle
(Ljapunov instability)
Flight directions:
The probability density of flight angles phi should be flat: no direction
is preferred.
p(vx):
The probability density of one (any!) velocity component.
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