Chaos in a very simple system:
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 v_x, v_y, 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 10^12), 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. Be patient,
it will be discussed when you are ready.
