The “Game” of Billiards

By Anna Rapoport

Boltzmann’s Hypothesis – a Conjecture for centuries ?

Boltzmann’s Ergodic Hypothesis (1870): For large systems of interacting particles in equilibrium time averages are close to the ensemble, or equilibrium, average.

The gas of hard balls is a classical model in statistical physics.

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.

Definition of Ergodicity

Take X to be a phase space, T – the evolution of the phase space, A – an invariant set under this evolution. Consider to be a measurement (a function on the phase space).

Ex: Map Ex: Map x →x → x + a mod 1 x + a mod 1 with with aaR\Q R\Q and the Lebesgue measure is ergodic.and the Lebesgue measure is ergodic.

Birkhoff Ergodic Theorem

Trick

“Boltzmann problem”: N balls in some reservoir

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

Mechanical Model

Billiards Billiard is a dynamical

system describing the motion of a point particle in a connected, compact domain Q RR22 or T2, with a piecewise Ck-smooth (k>2) boundary with elastic collisions from it.

V

QQ

+Q

0

Q -

Dispersing component

Q +

Focusing component

Q -

More Formally

D RR2 or T2 is a compact domain – configuration space; S is a boundary, consists of Ck curves:

Singular set:

Particle has coordinate q=(q1,q2) D and velocity v=(v1,v2) RR2

Inside D

m=1p=v p=constH=1/2

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

Set ||p||:=1; P’=DS1 is a phase space; t:P’ → P’ is a billiard flow;

By natural cross-sections reduce flow to map Cross-section – line transversal to a flow

dim P = 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

Integrability

Classical LiouvilleTheorem (mid 19C): in Hamiltonian system of 2 d.o.f. generalized coordinates: (q1,q2) conjugate momenta: (p1,p2)

If there are 2 conserved quantities K,H, s.t.

Another interpretation: The phase space is foliated by less than 2-dimensional invariant hypersurfaces (i.e. lines or points).

Ergodicity

Remind: The billiard is ergodic if the phase space cannot be decomposed into a sum of two regions of positive measure.

Geometric interpretation: A billiard in configuration space M is ergodic if almost all of its trajectories pass arbitrarily close to any given point of M with a direction arbitrarily close to any prescribed direction.

Caustics

Def: CAUSTICS – curves for which tangent billiard trajectories remains tangent after successive reflections.

Ex: Ellipses have conformal conics as caustics: ellipses and hyperbolas.

If one can find caustics, hence the system is non-ergodic.

Integrable billiards

Ellipse, circle, square. Any classical ellipsoidal billiard is integrable

(Birkhoff). Conjecture (Birkhoff-Poritski): Any 2-

dimensional integrable smooth, convex billiard is an ellipse..

2001 – Delshams et el showed that the Conjecture is locally true (under symmetric entire perturbation the ellipsoidal billiard becomes non-integrable).

Convex billiards

In 1976 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.

The billiard cannot be ergodic since N has a positive measure. Away from N the dynamics might be quite different.

Smoothness!!! The first convex billiard, which is ergodic (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

Demonstration of Chaotic Behavior

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. Later was proved dispersing billiards are ergodic.

In 2003 Simáni and Szász showed that the system of N balls which is reduced to a billiard in 2N x d dimension is ergodic.

Dispersing component

Q +

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 a `chaotic sea’. Examples: 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 (after Bunimovich)

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 D.Turaev considered the effect of smoothing of potential of dispersing billiards. In 2-dim it can give rise to elliptic islands.

The End