The “ Game ” of Billiards

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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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### 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.
• Boltzmann’s 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.
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 → ∞:
• Birkhoff proved that this convergence is a.e.
• Remind: the system is ergodic if for every A, (A) =0 or 1.
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 T2or T3 is ergodic for any N 2.
• No large N is assumed!
• 1970, Sinai verified this conjecture for N=2 on T2.
Trick

“Boltzmann problem”:N balls in some reservoir

“Billiard problem”:1 ball in higher dimensional phase space

Constants of motion
• Note that the kinetic energy is constant (set H=1/2)
• If the reservoir is a torus T3 (no collisions with a boundary) then also the total momentum is conserved (set P=0)
• Also assume B=0:

Dispersing component

Focusing component

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 (k>2) boundary with elastic collisions from it (def from Szácz).
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

m=1p=v

Reflection
• The angle of incidence is equal to the angle of reflection – elastic collision.
• The incidence angle [-/2;/2];
•  = /2corresponds to tangent trajectories
Phase Space
• H is preserved  ||p||:=1;
• P’=DSd-1is a phase space;
•  t:P’ → P’ isa 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

Statistical Properties
• Invariant measure under the billiard flow:
• CLT:
• Decay of correlation: ((n)~e-n, (n)~n-)
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.
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
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.
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).
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.
• 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 << l, a -- a near integrable system.
• b =a/2 -- ergodic

Dispersing component

Dispersing Billiards
• If all the components of the boundary are dispersing, the billiard is said to bedispersing. 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

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.

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.
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