Multi Dimensional Billiard-like Potentials Ph.D. Final Report by Anna Rapoport Under the Supervision of Prof. Vered Rom-Kedar Weizmann Institute of Science Rehovot, Israel December 27, 2007

Introduction  The Boltzmann’s Ergodic Hypothesis (1870): The LARGE system of gas molecules interacting in a box is ergodic.  The Boltzmann-Sinai Ergodic Hypothesis (1963): The system of N hard balls given on T 2 or T 3 is ergodic for any N≥2.  No large N is assumed!  Sinai verified this conjecture for N=2 on T 2 in  Better description of gas: Elastic collisions are replaced by interaction via smooth potential.  Donnay(Don96): The system of 2 particles with a finite range potential on a T 2 is non-ergodic.  Reduction to the billiard: The motion of N rigid d-dim balls in a d-dim box (d=2,3) corresponds to a billiard problem in a complicated n-dim domain, where n=2Nd.  Rom-Kedar and Turaev (RKTu98, TuRK03): Dispersing billiards with a smooth potential in 2-dim possess elliptic islands.

Hyperbolicity vs. ellipticity results  “Poincare conjecture”: periodic solutions are dense in a generic Hamiltonian systems.  Generic smooth Hamiltonians are neither ergodic nor integrable (i.e. possess a mixed phase space) MaMe74.  A C 1 –generic symplectic Diff is either Anosov (uniformly hyperbolic) or it has dense 1-elliptic points Newh77.  “Poincare conjecture” for C 1 case was established in PuRo83.  If a C 1 –generic symplectic Diff is not partially hyperbolic, then it has an elliptic periodic point SaXi06  In BaTo05 it was shown that soft multi-dimensional billiards with spherically symmetric, disjoint, finite- range potentials are hyperbolic.  We consider C r –generic symplectic Diff.

Main findings  Approximation of multi-dimensional Hamiltonian flows by billiards A. Rapoport, V. Rom-Kedar, and D. Turaev, “Approximating multi- dimensional Hamiltonian flows by billiards”, to appear in CMP (2006).  Non-ergodicity of the motion in n-dimensional (n>2) steep repelling dispersing potentials A. Rapoport and V. Rom-Kedar, “Nonergodicity of the motion in three- dimensional steep repelling dispersing potentials”, Chaos 16 (2006), no. 4 A. Rapoport, V. Rom-Kedar, and D. Turaev, “High dimensional linearly stable orbits in steep, smooth, strictly dispersing billiard potentials”, preprint (2007).  Corners effect on chaotic scattering in two dimensions A. Rapoport and V. Rom-Kedar, Corners effect on chaotic scattering in two dimensions, preprint (2007).

Billiards vs. Smooth Hamiltonian flow  Billiard flow:  Smooth Hamiltonian flow:

Approximation of multi-dimensional Hamiltonian flows by billiards:  We provide sufficient conditions for a reflection of a smooth Hamiltonian flow to converge to a reflection of a billiard flow. We prove that this convergence is smooth (continuous), where the billiard reflection is smooth (continuous) (extension of TuRk98). This gives a powerful theoretical tool for proving the persistence of various billiard trajectories in the smooth systems, and vice versa.  We define auxiliary billiard domains that asymptote, as  0 to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C r norm and an iteration scheme for improving this approximation.

Examples and Application:  Since the smooth flow and the billiard flow do not match in a boundary layer, we specify the width of it and the time spent in it. We also supply the estimates for the boundary layer width and the accuracy of the auxiliary billiard approximation for some typical potentials:  Applying these results to the billiards studied in DFRR01, we prove that the motion in steep potentials in various deformed ellipsoids (  controls the deformation) are non- integrable for an open interval of the steepness parameter

Non-ergodicity of the motion in 3-dimensional steep repelling dispersing potentials Simple symmetric billiard Proper rescaling allows to integrate numerically for arbitrarily small . The boundaries of the wedges could be found numerically by a continuation scheme Configuration space: Parameter space: Phase space (=0.04):

Non-ergodicity of the motion in n-dimensional steep repelling dispersing potentials  Theorem: There exist families of smooth billiard potentials which limit as the steepness parameter 0 to Sinai billiards in n-dimensional compact domains, yet, for arbitrary small  the corresponding smooth Hamiltonian flows have linearly stable (elliptic) periodic orbits.  Conjecture: Consider the family of Sinai billiards in n-dimensional space and the family of C r+1 smooth billiard potentials (with r>6) which limits, as the steepness parameter  0 to these billiards. Then, for any finite n, there exists an open set (in the C 2 topology) of such smooth billiard potentials, which, for arbitrary small , has an effective stability region near some periodic motion and is thus non-ergodic.

Geometry Potential

Non-dimensionalization  Geometrical parameters (l,r,L,R) and the steepness parameter  may be expressed (up to rescaling by r) by scaled non- dimensional parameters  The limit r is equivalent the limit 0.  =0 corresponds to a cusp; =1 corresponds n spheres collapse into one.  d – dimensionless length of the corner polygon;

One-dimensional periodic orbit Independent of n and R!

Stability of the periodic orbit The only place where n appears

Gaussian potentialCoulomb potential Analytic curve

Analytic bifurcation curves for the power-law potential near  =1.

Analytic bifurcation curves for the power-law potential near  =0.

Agreement with numerics: Re( ) At =0:At =0.99: Furthermore, calculating from (0.10) the origin of the second stability zone at =0 we obtain:

Limit n   vs.  0  We see that the limits n and 0 do not commute. For n  ,n (t) is unstable. For fixed n and 0 the periodic potential a(t;,n,,d,R,h) becomes concentrated on small intervals with sharp positive and negative spikes  the stability becomes a delicate issue.

Large  values  At = max (,d) the periodic orbit becomes a fixed point:  The stability of the diagonal orbit may be found analytically for   max (,d,r) -. The periodic orbit is well approximated by small oscillations near the fixed point  Hence the stability of this orbit is governed by the sign of  Defining We see that for n n c they are unstable. For they are always unstable

Phase space n=10 CoulombGaussian Projection to ||q||, ||p|| space shows no escape

Chaotic scattering in 2d  The scattering process: a ray of trajectories enters the interaction region, interacts with a scatterer and leaves the region. The common characteristics observed upon leaving the interaction region are the escape angle and the residence time.  Previous works:  Non-intersecting hard-wall circles ( Eckh87,GaRi89 );  Number of potential hills. Main question – mechanism of the transition from regular to chaotic scattering :  reflecting from barriers formed by potential hills (BOttG89, TrSmi89, BGOtt90,DGOY90,OttTel93,BKT94);  deflection induced by the topography(DVY93,DVY97).

Corners effect on chaotic scattering in 2d  We investigate the following 2 questions: Effect of a number of corners on the chaotic scattering for the billiard case. Corner’s effect on the chaotic scattering for a smooth. Geometry:

Potential effect for one corner geometry  We consider the exponential potential exp(-Q/)  We fix the corner geometry at =0.9 and find the interval of stability.  = is located inside the stability interval  = very close to period-doubling boundary  The sticky orbit - black

Parameterization  Interaction region:  (t) – the sticky (black) trajectory (=0.1403)  t e – the time (t) spends inside the interaction region.  Define the initial data:  Scattering function: x=y

Scattering function for the billiard case Counting-box dimension:

Scattering function for the smooth potential case No corners case: 1 corner case:

Self similarity of the scattering function IslandPeriod-doublingAbove SIBelow SI Billiard Small  value

The transition from regular to chaotic scattering  For  sc  max the scattering is regular.  Via the saddle-center bifurcation we move to the stability interval  pd  sc where the scattering is chaotic. Moreover, the scattering function shows very singular behavior (without well-pronounced smooth regions) predicted by LFiOtt91.  Further decreasing  leads through the period- doubling bifurcation to hyperbolic chaos. For 0 pd the scattering function possesses obvious self-similarity and approaches the scattering function corresponding to the billiard with two circles.

Conclusions:  The above works brought some understanding of important properties of multi-dimensional billiard like potentials and their relation to the corresponding hard-wall billiards.  We have shown that for the regular trajectories of the billiard those two systems are “close” and have found how “close” they are. This gives a tool to understand properties of the smooth Hamiltonian system knowing the properties of the corresponding billiard.  On the other hand we presented the case when the smooth system behaves completely different from the billiard: We proved that unlike the dispersive billiards, which are ergodic, smooth billiard potential systems possess islands of stability associated with periodic orbits going to a corner.  Finally we studied the corners effect on the scattering function in the two-dimensional case. Hopefully these results will contribute to better understanding of the Boltzmann Ergodic Hypothesis being an open problem for many years.

Bibliography  V.J. Donnay Erg.Th.& Dyn. Sys. 16, (1996)  V.Rom-Kedar and D. Turaev Nonlinearity 11 no.3, (1998)  D. Turaev and V. Rom-Kedar J.Stat. Phys. 112, no. 3-4, (2003)  P.Balint and I. P. Toth, Discrete Contin. Dyn. Syst. 15, no. 1, 37–59 (2006)  C. Pugh and R. C. Robinson, Ergod. Th. Dynam. Sys. 3, (1983).  A. Delshams, Yu. Fedorov, and R. Ram Í rez-Ros, Nonlinearity 14, no. 5, 1141–1195, (2001)  Y.T. Lau, J.M. Finn, and E. Ott, Phys.Rev.Lett. 66 (1991).