The Game of Billiards

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The Game of Billiards

• By Anna Rapoport (from my proposal)

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Boltzmanns Hypothesis a Conjecture for
centuries?

• The gas of hard balls is a classical model in
  statistical physics.
• Boltzmanns Ergodic Hypothesis (1870) For large
  systems of interacting particles in equilibrium
  time averages are close to the ensemble, or
  equilibrium average.
• Let ? is a measurement, a function on a phase
  space, equilibrium measure µ, and let f be a
  time evolution of a phase space point.
• One should define in which sense it converges.
• It took time until the mathematical object of the
  EH was found.

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The First Mean Ergodic Theorem

• In 1932 von Neumann proved the first ergodic
  thorem
• Let M be an abstract space (the phase space) with
  a probability measure µ, f M ? M is a measure
  preserving transformation (?(f -1(A) ) ?(A) for
  any measurable A), ? ? L2(?),as n ? 8
• Birkhoff proved that this convergence is a.e.
• Remind the system is ergodic if for every A,
  ?(A) 0 or 1.

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From Neumann to Sinai (1931-1970)

• 1938-39, Hedlung and Hopf found a method for
  demonstrating the ergodicity of geodesic flows on
  compact manifolds of negative curvature. They
  have shown that here hyperbolicity implies
  ergodicity.
• 1942, Krylov discovered that the system of hard
  balls show the similar instability.
• 1963, The Boltzmann-Sinai Ergodic Hypothesis The
  system of N hard balls given on T2 or T3 is
  ergodic for any N? 2.
• No large N is assumed!
• 1970, Sinai verified this conjecture for N2 on
  T2.

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Trick
Boltzmann problem N balls in some reservoir
Billiard problem 1 ball in higher dimensional
phase space
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Mechanical Model
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Constants of motion

• Note that the kinetic energy is constant (set
  H1/2)
• If the reservoir is a torus T3 (no collisions
  with a boundary) then also the total momentum is
  conserved (set P0)
• Also assume B0

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Billiards

• Billiard is a dynamical system describing the
  motion of a point particle in a connected,
  compact domain Q ? Rd or Td, with a piecewise
  Ck-smooth (k2) boundary with elastic collisions
  from it (def from Szácz).

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More Formally

• D ? Rd or Td (d 2) is a compact domain
  configuration space
• S is a boundary, consists of Ck (d-1)-dim
  submanifolds
• Singular set
• Particle has coordinate q(q1,,qd)? D and
  velocity v(v1,,vd)? Rd

Inside D
m1??pv
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Reflection

• The angle of incidence is equal to the angle of
  reflection elastic collision.

• The incidence angle ???-?/2?/2
• ?? ??/2?corresponds to tangent trajectories

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Phase Space

• H is preserved ? p1
• PD?Sd-1 is a phase space
• ? tP ? P is a billiard flow
• By natural cross-sections reduce flow to map
• Cross-section hypersurface transversal to a
  flow
• dim P (2d-2) and P? P (It consists of all
  possible outgoing velocity vectors resulting from
  reflections at S. Clearly, any trajectory of the
  flow crosses the surface P every time it
  reflects.)
• This defines the Poincaré return map

T - billiard map
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Singularities of Billiard Map
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Statistical Properties

• Invariant measure under the billiard flow
• CLT
• Decay of correlation
  (?(n)e-?n, ?(n)n-?)

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A little bit of History

• For Anosov diffeomorfisms Sinai, Ruelle and Bowen
  proved the CLT in 70th, at the same time the exp.
  decay of correlation was established.
• 80th Bunimovich, Sinai, Chernov proved CLT for
  chaotic billiards recently Young, Chernov showed
  that the correlation decay is exponential.
• It finally becomes clear that for the purpose of
  physical applications, chaotic billiards behave
  just like Anosov diffeomorphisms.

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Lyapunov exponents indicator of chaos in the
system

• If the curvature of every boundary component is
  bounded, then Oseledec theorem guarantees the
  existence of 2d-2 Lyapunov exponents at a.e.
  point of P.
• Moreover their sum vanishes

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Integrability

• Classical LiouvilleTheorem (mid 19C) in
  Hamiltonian dynamics of finite N d.o.f.
  generalized coordinates conjugate momenta
  Poisson brackets If
  conserved quantities Kj as many d.o.f.N are
  found system can be reduced
  to action-angle variables by quadrature only.

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Integrable billiards

• Ellipse, circle.
• Any classical ellipsoidal billiard is integrable
  (Birkhoff).
• Conjecture (Birkhoff-Poritski) Any 2-dimensional
  integrable smooth, convex billiard is an ellipse.
• Veselov (91) generalized this conjecture to
  n-dim.
• Delshams et el showed that the Conjecture is
  locally true (under symmetric entire perturbation
  the ellipsoidal billiard becomes non-integrable).

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Convex billiards

• In 1973 Lazutkin proved if D is a strictly
  convex domain (the curvature of the boundary
  never vanishes) with sufficiently smooth
  boundary, then there exists a positive measure
  set N?P that is foliated by invariant curves (he
  demanded 553 deriv., Douady proved for 6(conj.
  4)).
• The billiard cannot be ergodic since N has a
  positive measure. The Lyapunov exponents for
  points x?N are zero. Away from N the dynamics
  might be quite different.
• Smoothness!!! The first convex billiard, which is
  ergodic and hyperbolic (its boundary C1 not C2)
  is a Bunimovich stadium.

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Stadium-like billiards

• A closed domain Q with the boundary consisting
  of two focusing curves.
• Mechanism of chaos after reflection the narrow
  beam of trajectories is defocused before the next
  reflection (defocusing mechanism, proved also in
  d-dim).
• Billiard dynamics determined by the parameter b
• b
• b a/2 -- ergodic

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Dispersing Billiards

• If all the components of the boundary are
  dispersing, the billiard is said to be
  dispersing. If there are dispersing and neutral
  components, the billiard is said to be
  semi-dispersing.
• Sinai introduced them in 1970, proved (2 disks on
  2 torus) that 2-d dispersing billiards are
  ergodic. In 1987 Sinai and Chernov proved it for
  higher dimensions (2 balls on d torus).
• The motion of more than 2 balls on Td is already
  semi-dispersing. 1999 Simáni and Szász showed
  that N balls on Td system is completely
  hyperbolic, countable number of ergodic
  components, they are of positive measure and
  K-mixing.
• 2003 they showed that the system is B-mixing.

Try to play
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Generic Hamiltonian

• Theorem (Markus, Meyer 1974) In the space of
  smooth Hamiltonians
• The nonergodic ones form a dense subset
• The nonintegrable ones form a dense subset.

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.
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Billiards with a mixed PS

• The mushroom billiard was suggested by
  Bunimovich. It provides continuous transition
  from chaotic stadium billiard to completely
  integrable circle billiard. The system also
  exhibit easily localized chaotic sea and island
  of stability.

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Mechanisms of Chaos
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
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Adding Smooth Potential

• High pressure and low temperature the hard
  sphere model is a poor predictor of gas
  properties.
• Elastic collisions could be replaced by
  interaction via smooth potential.
• Donnay examined the case of two particles with a
  finite range potential on a T2 and obtained
  stable elliptic periodic orbit non-ergodic.
• V.Rom-Kedar and Turaev considered the effect of
  smoothing of potential of dispersing billiards.
  In 2-dim it can give rise to elliptic islands.

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Current Results

• Generalization of billiard-like potential to
  d-dim.
• Conditions for smooth convergence of a smooth
  Hamiltonian flow to a singular billiard flow.
• Convergence Theorem is proved.

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Research Plan

• Consider one of the 3-dim billiards built by Nir
  Davidsons group. Investigate its ergodic
  properties, study phase space.
• Find a mechanism which gives an elliptic
  point of a Poincaré map of a smooth
  Hamiltonian system (d-dim) (multiple
  tangency, corner going trajectories)
• Find whether the return map is non-linearly
  stable, so that KAM applies.
• The resonances will naturally arise.