The “ Game ” of Billiards By Anna Rapoport (from my proposal) Boltzmann ’ s Hypothesis – a Conjecture for centuries? The gas of hard balls is a classical model in statistical physics.
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By Anna Rapoport (from my proposal)
“Boltzmann problem”:N balls in some reservoir
“Billiard problem”:1 ball in higher dimensional phase space
By natural cross-sections reduce flow to map
T - billiard map
Try to play
The Generic Hamiltonian possesses a mixed phase space. The islands of stability (KAM islands) are situated in `chaotic sea’. Examples: cardioid, non-elliptic convex billiards, mushroom.
Integrabiliy (Ellipse) - divergence and convergence of neighboring orbits are balanced
Defocusing (Stadium) - divergence of neighboring orbits (in average) prevails over convergence
Dispersing (Sinai billiards) - neighboring orbits diverge