Ergodicity of classical billiard balls

Physica A 194 (1993) North-Holland

86-92

Ergodicity Domokos

of classical billiard balls *

Sz6sz

Mathematical Institute of the Hungarian Academy of Sciences, P.O. B. 127, H-1364 Budapest, Hungary

Dedicated to Elliott Lieb on the occasion of his 60th birthday Sinai, in 1970, proved the ergodicity of two discs on the 2-torus. Further essential progress was only reached in 1987, when Chernov and Sinai established the ergodicity of two billiard balls on the v-torus. Then, by applying a strategy suggested by Kriimli, Siminyi and myself in 1989, we obtained the ergodicity of three and four balls on the v-torus (v 3 3 in the latter case) in 1991 and 1992, while recently Simanyi proved that of N balls on the v-torus whenever v 2 N. After a survey of this progress the study of toric billiards with cylindric scatterers is initiated and the K-property of a general class of these is claimed.

1. The ergodic hypothesis

and hyperbolic

billiards

Both the verification of Boltzmann’s ergodic hypothesis and the understanding of the mixing behaviour of physical systems are substantial questions of the foundations of statistical physics. In 1942, a Russian physicist, N.S. Krylov [l] suggested that underlying the aforementioned properties should be the exponential instability discovered earlier by Hedlund and Hopf as the “cause” of ergodicity in a model dynamical system (geodesic flow in a fundamental domain of the Bolyai-Lobachevsky plane). His manifestation of hyperbolicity in a “perfect gas”, i.e. in a system of elastic hard balls on one hand and the development in the “60”s of the theory of hyperbolic dynamical systems (Smale, Anosov) on the other hand, influenced Sinai [2] (i) to initiate the theory of hyperbolic dynamical systems with singularities, (ii) to formulate his version of Boltzmann’s ergodic hypothesis (claiming that systems of a finite (~2) number of classical billiard balls given on the 2- or 3-torus are ergodic on the submanifold of the state space specified by the trivial integrals of motion) and (iii) to prove the first remarkable result of the theory: the system of two * Research 1902.

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D. Szctsz I Ergodicity of classicalbilliards balls

87

elastic discs on the 2-torus is ergodic, and is even Kolmogorov mixing (we note that this system is a particular case of a Sinai billiard). Assume that, in general, a system of N (~2) balls of radii r > 0 are given on T”, the v-dimensional unit torus (V 2 2). Denote the phase point of the ith ball by ( qi, vi) ET” x OX”.The configuration space Q of the N balls is a subset of TN’“: from TN’” we cut out ( t ) cylindric scatterers: ej.j={Q=(ql

,...,

qN)~TN’“:Iq;-q,)<2r},

1s i < j c N. The energy H = 1 Cr uy and the total momentum P = CF ui are first integrals of motion. Thus, without loss of generality, we can assume that H = $ and P = 0 and, moreover, that the sum of spatial components B = Cy qi = 0 (if P # 0, then the center of mass has an additional conditionally periodic or periodic motion). For these values of H, P and B, the phase space of the system reduces to M : = Q x YPN,v_v_ 1 where qi=O}

Q:={Q& 1

with d : = dim Q = N. Y - v, and further 9, denotes, in general, the k-dimensional unit sphere. It is easy to see that the dynamics of the N balls, determined by their uniform motion with elastic collisions on one hand, and the billiardflow {S’: t E R} in Q with specular reflections at 82 on the other hand, conserve the Liouville measure dp = const x dq dv and are isomorphic.

In the aforementioned billiards the smooth components aQi of the boundary of Q are concave from inside the domain and this concavity generates their hyperbolicity. We can, in general, consider billiards in bounded, connected, closed domains on the d-torus Td. We say that such a billiard is semi-dispersing (dispersing) if the smooth components of the boundary are concave (strictly concave) from inside the domain. Thus the system of two hard balls is dispersing while that of N 2 3 balls is semi-dispersing. Now the underlying connection with the theory of hyperbolic dynamical systems is that dispersing billiards are manifestly hyperbolic while in the case of semi-dispersing billiards hyperbolicity is still present but it is somewhat hidden. It is, however, easy to see that billiards are not smooth everywhere; in fact, they are discontinuous on a finite union of codimension one submanifolds of the phase space corresponding to orbits that are either tangent to a smooth component aQi or hit the intersection of two (or more) smooth boundary components. The technical consequence of this discontinuity is that smooth components of the local stable and unstable manifolds, though they can be shown to exist almost everywhere, can be arbitrarily small; indeed, this is the basic additional difficulty in the theory of hyperbolic dynamical systems with discontinuities.

88

2. Chernov-Sinai

D. Szrisz I Ergodicity of classical billiards balls

theory of local ergodicity for semi-dispersing

billiards

For detecting the hyperbolicity in a semi-dispersing billiard, the basic notion is the sz.@ciency of a trajectory or a segment of it. For simplicity we formulate it for a regular trajectory segment S ‘a,bl~ avoiding discontinuities. Let Sax = (Q, V) E M and consider the hyperplanar wavefront y,(S”x) : = {(Q + dQ, V): dQ small E Rd and (de, V) = 0) (by denoting Z-(X)= Q for x = (Q, V) we see that, indeed, n( yO) is part of a hyperplane). We say that the trajectory segment S “,“x is sz4fJicient if 7r(Sb-y,,) is strictly convex. (To obtain a geometric or optical feeling of this notion, the reader is suggested to imagine mirror-surfaced scatterers.) A phase point x EM is sufficient if its trajectory is sufficient (i.e. it contains a sufficient trajectory segment). In physical terms, sufficiency of a trajectory segment means that, during the time interval [a, b], the trajectory of x encounters all degrees of freedom of the system. If a trajectory segment is not sufficient, then the curvature of rr(Sby,,) at rr(S’x) necessarily vanishes in certain directions forming the so-called neutral subspace. Simple geometric considerations (cf. ref. [3]) show that a sufficient trajectory segment generates an expansion rate uniformly larger than 1 in some neighbourhood of the point S”x. Then, by Poincare recurrence and the ergodic theorem, it is not hard to see that, in some neighborhood of Sax, the Lyapunov exponents of the system are not zero. In other words, in this neighborhood, the system is hyperbolic. This observation should motivate the non-trivial Fundamental theorem for semi-dispersing billiards [4,5]. Assume that a semidispersing billiard satisfies some geometric conditions and the Chernov-Sinai ansatz, a condition strongly connected with the singularities of the system. If x E M is a sufficient point, then it has an open neighborhood U in the phase space belonging to one ergonent (i.e. ergodic component).

The property expressed in the statement is usually called local ergodicity. If almost every phase point of a semi-dispersing billiard is sufficient, then, of course, it may have at most a countable number of ergonents. In some cases it is not hard then to derive the global ergodicity of the system, i.e. to show that there is just one ergonent in the phase space. It also follows from the general theory that, on each ergonent, the system is Kolmogorov mixing. A much important consequence is thus the following Corollary [4,5]. Every dispersing billiard is ergodic, and, moreover, is a K-flow. In particular, the system of N = 2 balls on the v-torus is a K-flow if r<1/2.

D. Snisz

I Ergodicity of classical billiards balls

3. Global ergodicity of semi-dispersing

89

billiards and of classical billiard balls

Assume that, for a semi-dispersing billiard, we can verify the Chernov-Sinai ansatz. Then the fundamental theorem suggests that, to obtain global ergodicity of the system, one should show that the set of non-sufficient points is a topologically small set in a sense ensuring that it cannot separate different open ergodic components. This idea was made rigorous in the study of a paradigma model of a billiard on the 3-torus with two cylindric scatterers by A. Krimli, N. Simanyi and myself [6]. The main conclusions of this work have been the following: (1) The appropriate notion of topological smallness of a subset is strongly related to having (topological) codimension two. More precisely, we say that a subset A of the phase space is residual if it can be covered by a countable union of codimension-two closed sets of measure zero. Then, indeed, the complement of a residual subset certainly contains an arcwise connected G, subset of full measure. (2) Important novel technical tools for showing that non-sufficient points form a residual subset in the phase space are the ball-avoiding theorems. As an illustration we present the first, simplest version of such a result. Ball-avoiding theorem [6]. Assume we are given a dispersing billiard, an open

subset G in its phase space, and H C Iw such that inf H = -CC and sup H = ~0. Then the set A,(G)

:= {x E M: S’xFG

whenever t E H}

is residual. We note that the statement p(A,(G)) = 0 easily follows from the mixing property of the dispersing billiard in question. Let us focus now our attention to hard ball systems. The task as before is to show that non-sufficient points form a residual subset. A natural approach to cope with this problem is the following: If one assumes that a trajectory segment is rich in the sense that it contains a reasonable amount of collisions of different pairs of particles, then it is natural to expect that, at least typically, it is sufficient. The first success of this approach was the proof of ergodicity of three billiard balls obtained by Kramli, Siminyi and myself [7]. This proof suggested a strategy for establishing the ergodicity of any number of balls consisting of the following main steps. (1) Find an appropriate concept of richness of a trajectory or a finite trajectory segment. The definition need not be given in dynamical terms but

90

D. Szcisz I Ergodicity of classical billiards balls

rather

in terms of the symbolic

noting torus. balls,

that

the definition

To provide

a feeling

the appropriate

at least different different,

three,

of richness,

definition

then

collisions minimal

collision

of richness

sequence may

Also, it is worth

on the dimension

of the

we recall that, in the case of the three

is the following:

a rich trajectory

of the orbit.

depend

segment

if the dimension

of the torus is

should

at least

contain

three

(in time order). For this there are only two, essentially possibilities, namely {1,2}, {2,3}, {1,2} or {1,2}, {2,3},

{ 1,3} meaning - in the first case - that there is first one (or several) collision of balls 1 and 2, followed by one (or several) collision of balls 2 and 3, and then again balls 1 and 2 collide. In the two-dimensional existence of at least four such islands of collisions.

case, richness

requires

the

(2) Geometric-algebraic considerations. With some simplification one can say that, in a sufficiently small neighborhood of a phase point x with a rich trajectory segment, non-sufficient points are solutions of certain implicitly given equations. As a result, in the treatable cases at least, one can prove that non-sufficient points form subsets of certain submanifolds of codimension at least two. This is, of course, amply suitable for our purposes. This part of the proof uses geometric and algebraic considerations. (3) Prove that non-rich phase points form a codimension-two subset. This statement can be derived from more sophisticated versions of the ball-avoiding theorem. The technique of the proof is basically an improved version of Hopf’s classical method, a fundamental tool for establishing local or global ergodicity of hyperbolic dynamical systems (cf. refs. [2-51). (4) Singular orbits and the Chernov-Sinai ansatz. In steps 2 and 3, it is convenient to first restrict the arguments to neighborhoods of non-singular points. The treatment of points with a single singularity and the verification of the Chernov-Sinai ansatz are closely related since both concern sufficient points with exactly one singularity on their trajectories. By applying this strategy, the following further results have been obtained: (1) Kramli, Simanyi and myself [S] established the K-property of four billiard balls on the v-torus whenever v 2 3. (2) Bunimovich, Liverani, Pellegrinotti and Sukhov [9] found a model of hard balls with an arbitrary number of particles in an appropriately chosen domain, which they could prove the K-property for. (3) Simanyi [lo] proved the K-property of N balls on the v-torus whenever v 2 N. For step 3, he found a nice and efficient “uniform” ball avoiding theorem. For step 2 - the only place where the condition on the dimension appears! - his approach is based on a beautiful characterization of the neutral subspace: the connected path formula.

D. Srhz

4. Concluding

I Ergodicity of classicalbilliardsballs

91

remarks

1. As shown in section 1, systems of classical billiard balls are isomorphic to certain semi-dispersing billiards with cylindric scatterers. Here the configuration of scatterers reflects the permutation symmetry of the balls. Thus a more general model is a billiard with cylindric scatterers. John Mather asked whether a condition that, for instance, the intersection of the constituent subspaces of these cylinders is trivial, is sufficient to ensure the ergodicity of the corresponding billiard. I expect that the answer is yes if, in addition, we exclude the existence of trivial integrals of motion. To illustrate what I mean, consider the billiard on the 4-torus with two cylindric scatterers: C, : = {Q E T4: 1q: + qz[ =Z r} and C,:= {QET4: s r}. Here beside the total energy, of course, td+d IJ: + uz and u: + V: are also conserved. The positive general result that I can prove is the following [ll]: Assume we are given a family C,,: 1
References [l] N.S. Krylov, Works on Foundations of Statistical (Princeton Univ. Press, Princeton, 1979). (21 Ya.G. Sinai, Usp. Mat. Nauk 25 (1970) 141.

Physics,

Princeton

Series in Physics

92

[3] [4] [5] [6] [7] [S] [9] [IO] [ll] [12] [13] [14] [15] [16] [17]

D. Szcisz I Ergodicity of classical billiards balls

L.A. Bunimovich and Ya.G. Sinai, Mat. Sbornik 90 (1973) 415. Ya.G. Sinai and N.I. Chernov, Usp. Mat. Nauk 42 (1987) 153. A. Kramli, N. Simanyi and D. Szlsz, Commun. Math. Phys. 129 (1990) 535. A. Kramli, N. Simlnyi and D. SzLsz, Nonlinearity 2 (1989) 311. A. Kramli, N. Simanyi and D. Szasz, Ann. Math. 133 (1991) 37. A. Kramli, N. Siminyi and D. Szasz, Commun. Math. Phys. 144 (1992) 107. L. Bunimovich, C. Liverani, A. Pellegrinotti and Yu. Sukhov, Commun. Math. Phys. (1992) 357. N. Simanyi, Invent. Math. 108 (1992) 521; and article II, to appear. D. Szasz, Toric billiards with orthogonal cyclindric scatterers, in preparation. G. Gallavotti, D. Ornstein, Commun. Math. Phys. 38 (1974) 83. L.A. Bunimovich, N.I. Chernov and Ya.G. Sinai, Usp. Mat. Nauk 46 (1991) 43. M. Wojtkowski, Commun. Math. Phys. 126 (1990) 507; 127 (1990) 425. L. Bunimovich, Commun. Math. Phys. 130 (1990) 599. D. Szasz, Commun. Math. Phys. 145 (1992) 595. M. Wojtkowski, Commun. Math. Phys. 105 (1986) 391.

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