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 T2 or 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

Mechanical Model

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:

Billiards Billiard is a dynamical

system describing the motion of a point particle in a connected, compact domain Q RRd or Td, with a piecewise Ck-smooth (k>2) boundary with elastic collisions from it (def from Szácz).

V

QQ

+Q

0

Q -

Dispersing component

Q +

Focusing component

Q -

More Formally

D RRd 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) RRd

Inside D

m=1p=v

Reflection

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

• The incidence angle [-/2;/2];

• = /2corresponds to tangent trajectories

Phase Space

H is preserved ||p||:=1; P’=DSd-1 is a phase space; t:P’ → 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

Singularities of 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 NP 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 xN 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.

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 << l, a -- a near integrable

system. b =a/2 -- ergodic

n + 1Q

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.

Dispersing component

Q +

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

Defocusing (Stadium) - divergence of neighboring orbits (in average) prevails over convergence

Dispersing (Sinai billiards) - neighboring orbits diverge

Integrabiliy (Ellipse) - divergence and convergence of neighboring orbits are balanced

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.

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.

Research Plan

Consider one of the 3-dim billiards built by Nir Davidson’s 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.