We consider the motion of two point masses along a vertical half-line that are subject to constant gravitational force and collide elastically with each other and the floor. This model was introduced by Wojtkowski who established hyperbolicity and ergodicity in case the lower ball is heavier. Here, we investigate the dynamics in discrete time and prove that, for an open set of the external parameter (the relative mass of the lower ball), the system mixes polynomially—modulo logarithmic factors, correlations decay as —and satisfies the Central Limit Theorem.
REFERENCES
1.
V.
Baladi
and S.
Gouëzel
, J. Mod. Dyn.
4
, 91
(2010
).2.
P.
Bálint
, N.
Chernov
, and D.
Dolgopyat
, Commun. Math. Phys.
308
, 479
(2011
).3.
P.
Bálint
and S.
Gouëzel
, Commun. Math. Phys.
263
, 461
(2006
).4.
P.
Bálint
and I.
Melbourne
, J. Stat. Phys.
133
, 435
(2008
).5.
P.
Bálint
and I. P.
Tóth
, Commun. Math. Phys.
243
, 55
(2003
).6.
P.
Bálint
and I. P.
Tóth
, Ann. Henri Poincare
9
, 1309
(2008
).7.
P.
Bálint
and I. P.
Tóth
, “Example for exponential growth of complexity in a finite horizon multi-dimensional dispersing billiard
,” preprint, available at http://www.math.bme.hu/∼pet/pub/ellenpelda_20120206.pdf.8.
L. A.
Bunimovich
, Ya. G.
Sinai
, and N. I.
Chernov
, Russ. Math. Surveys
46
, 47
(1991
).9.
J.-R.
Chazottes
and S.
Gouëzel
, “Optimal concentration inequalities for dynamical systems
,” preprint arXiv:1111.0849.10.
N.
Chernov
, J. Stat. Phys.
94
(3-4
), 513
(1999
).11.
N.
Chernov
, Russ. Math. Surveys
46
, 187
(1991
).12.
N.
Chernov
and D.
Dolgopyat
, Russ. Math. Surveys
64
, 651
(2009
).13.
N.
Chernov
and R.
Markarian
, “Chaotic billiards
,” Math. Surv. Monogr. Vol. 127 (American Mathematical Society
, Providence
, RI
, 2006
).14.
N.
Chernov
and R.
Markarian
, Commun. Math. Phys.
270
, 727
(2007
).15.
N.
Chernov
and L.-S.
Young
, “Decay of correlations for Lorentz gases and hard balls
,” in Hard Ball Systems and the Lorentz Gas
, Encyclopaedia of Mathematical Sciences, edited by D.
Szasz
(Springer
, New York
, 2000
), Vol. 101
, pp. 89
–120
.16.
N.
Chernov
and H.-K.
Zhang
, Nonlinearity
18
(4
), 1527
(2005
).17.
R.
Dorfmann
, A Introduction to Chaos in Non-Equilibrium Statistical Mechanics
(Cambridge University Press
, Cambridge, England
, 1999
).18.
S.
Gouëzel
, Probab. Theory Relat. Fields
, 128
, 82
(2004
).19.
S.
Gouëzel
(private communication, 2011).20.
A.
Katok
and J.-M.
Strelcyn
, Invariant Manifolds, Entropy and Billiards; Smooth Maps With Singularities
, Lecture Notes in Mathematics (Springer
, New York
, 1986
), Vol. 1222
.21.
Anomalous Transport: Foundations and Applications
, edited by R.
Klages
, G.
Radons
, and I. M.
Sokolov
(Wiley
, New York
, 2008
).22.
C.
Liverani
, B.
Saussol
, and S.
Vaienti
, Ergod. Theory Dyn. Syst.
19
, 671
(1999
).23.
C.
Liverani
and M. P.
Wojtkowski
, “Ergodicity in Hamiltonian systems
,” in Dynamics Reported
(1995
), Vol. 4
, pp. 130
–202
.24.
Y.
Pomeau
and P.
Manneville
, Commun. Math. Phys.
74
, 189
(1980
).25.
R.
Markarian
, Ergod. Theory Dyn. Syst.
24
, 177
(2004
).26.
27.
I.
Melbourne
, Trans. Am. Math. Soc.
359
, 2421
(2007
).28.
I.
Melbourne
and M.
Nicol
, Ann. Probab.
37
(2
), 478
(2009
).29.
I.
Melbourne
and D.
Tershiu
, “Decay of correlations for nonuniformly expanding systems with general return times
,” preprint, available at http://personal.maths.surrey.ac.uk/st/I.Melbourne/papers/upper.pdf.30.
I.
Melbourne
and A.
Török
, Isr. J. Math.
144
, 191
(2004
).31.
I.
Melbourne
and A.
Török
, “Convergence of moments for Axiom A and nonuniformly hyperbolic flows
,” Ergod. Theory Dyn. Syst.
32.
M.
Pollicott
and R.
Sharp
, Nonlinearity
22
, 2079
(2009
).33.
N.
Simányi
, Commun. Math. Phys.
182
(2
), 457
(1996
).34.
D.
Szász
and T.
Varjú
, J. Stat. Phys.
129
, 59
(2007
).35.
M. P.
Wojtkowski
, Commun. Math. Phys.
126
, 507
(1990
).36.
L.-S.
Young
, Ann. Math.
147
, 585
(1998
).37.
L.-S.
Young
, Isr. J. Math.
110
, 153
(1999
).© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
