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Statistical properties of the system of two falling balls

### Abstract

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.

© 2012 American Institute of Physics

Received 28 November 2011
Accepted 18 February 2012
Published online 20 June 2012

Lead Paragraph:
Dynamical systems with intermittency—when the evolution alternates between chaotic and regular patterns—are popular models for describing a wide range of physical phenomena and thus have been in the focus of active investigation for several decades.^{17,21,22,24} Of particular relevance are systems in which the chaotic component is strong enough to ensure ergodicity with respect to a natural invariant measure; nonetheless, there is a significant regular component which makes the dynamics only non-uniformly hyperbolic. This often results in slower—polynomial, in contrast to exponential—rates in the decay of correlations. Another question of high importance is that of statistical limit laws, in particular, whether a natural class of observables exhibits standard or non-standard limit theorems (in other words, normal vs. anomalous diffusion). Despite of their significance, the class of systems for which detailed and mathematically rigorously justified information is available on the above mechanism is quite limited. The best understood models are one dimensional maps with neutral fixed points.^{22,18,9,26,32} There is quite much known about rates of mixing and statistical limit laws in certain two dimensional billiard systems of physics origin, in particular stadia,^{16,3} Lorentz gases with infinite horizon,^{34,12} and dispersing billiards with cusps.^{14,2} In the present paper, we investigate another billiard type model with intermittent behavior. The system of two falling balls, introduced by Wojtkowski in Ref. 35, describes the motion of two point particles of mass and that move along the vertical half-line, subject to constant gravitational force, and collide elastically with each other and the floor. We consider the case when the lower ball is heavier (i.e., ) which corresponds to ergodic and hyperbolic dynamics, while the regular component of intermittency is related to arbitrary long series of bounces of the lower ball on the floor before hitting the upper ball. We present a detailed analysis of this model and prove that correlations decay, modulo logarithmic factors, as . This rate is summable, accordingly, the central limit theorem is also proved; that is, the system exhibits normal diffusion.

Acknowledgments:
We are grateful to Imre Péter Tóth for his encouragement and enlightening discussions. The work reported in this paper has been developed in the framework of the project “Talent care and cultivation in the scientific workshops of BME” project. This project is supported by the grant TÁMOP-4.2.2.B-10/1–2010-0009. Our research is also connected to the scientific program of the “Development of quality-oriented and harmonized R + D + I strategy and functional model at BME” project. This project is supported by the New Széchenyi Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002). Part of this work was done while P.B. and A.N.V visited the Fields Institute, Toronto and while G.B. and A.N.V. visited the MPIPKS Dresden; we thank both institutions for their hospitality and stimulating research atmosphere. The financial support of the Bolyai Scholarship of the Hungarian Academy of Sciences (P.B.), and OTKA (Hungarian National Fund for Scientific Research), grant K71693 (all authors) is thankfully acknowledged.

Article outline:

I. SETTING
A. Description of the dynamical system
1. Historical background
2. The dynamical system
3. Piecewise smoothness
4. Non-uniform hyperbolicity
5. The first return map and the singularity stripes
B. Statement of results and structure of proofs
1. Results
2. Structure of proofs
II. ANALYSIS OF THE FIRST RETURN SETS
A. Notations concerning the geometry of the phase space
B. Bounding functions
C. A Simplified model
D. Straightening the stripes
E. Overview
III. REGULARITY PROPERTIES OF THE FIRST RETURN MAP
A. Involution
B. The asymptotics of F
C. Hyperbolicity
D. Bounded curvature
E. Distortion bounds
F. Absolute continuity
G. Regularity of the roof function
IV. GROWTH LEMMA
A. The first iterate of *T*
B. The second iterate
1. Singularities of the second iterate
2. Growth lemma for the second iterate
V. OUTLOOK
A. Extension of the results to a larger set of mass parameters
B. Decay of correlations for the flow

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2012-06-20

2016-09-25

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