Statistical properties of the system of two falling balls

Transcription

1 Statistical properties of the system of two falling balls András Némedy Varga Joint work with: Péter Bálint Dr. 1 Gábor Borbély Budapest University of Technology and Economics 1 Department of Differential Equations Dresden

2 The system of two falling balls v 2 m 2 " " h 2 gravity v 1 m 1 h 1

3 Brief history For n balls (instead of just two) if m 1 > m 2 > > m n then the system is hyperbolic (Wojtkowski 1990). It is enough to assume that m 1 m 2 m n, but at least one inequality is strict (Simányi 1996). For m 1 = m 2 the system is integrable, and if m 1 < m 2 then elliptic periodic orbit exists. The system of two falling balls is ergodic iff m 1 > m 2 (Liverani-Wojtkowsi 1995).

4 Poincaré-section and Wojtkowski coordinates Let m := m 1 and set m 1 + m 2 = 1. Also set J := mgh mv (1 m)gh (1 m)v 2 2 = 1 2. The phase space M = { (h 1, v 1, h 2, v 2 ) 0 < h 1 < h 2, J = 1 2}.

5 Poincaré-section and Wojtkowski coordinates Let m := m 1 and set m 1 + m 2 = 1. Also set J := mgh mv (1 m)gh (1 m)v 2 2 = 1 2. The phase space M = { (h 1, v 1, h 2, v 2 ) 0 < h 1 < h 2, J = 1 2}. A Poincaré-section M := { (h 1, v 1, h 2, v 2 ) M h 1 = 0, v 1 > 0 }. In Wojtkowski coordinates h = 1 2 mv 1 2 and z = v 2 v 1 we have M = { (h, z) (0 < h < J) (J h > 1 2 (1 m)v 2 2)}. These coordinates are invariant between collisions.

6 Expressing the dynamics Balls won t collide M 2 M ( ) 1 0 DF 2 = 2 hm 1 Balls will collide M 1 M 1 2mαz DF 1 = 2 1 q1 2 2αz hm +αz2 m q 1 h m +αz2 det(df i ) = 1 hence Lebesgue-measure is invariant.

7 The phase space 2 z m h 1 2

8 Result Theorem There is an open interval I ( 1 2, 1) such that for all m I the system enjoys polynomial decay of correlations f (g T n )dµ fdµ gdµ C f,g log 3 (n) n 2 M M and satisfies the central limit theorem, for Hölder observables. M

9 First return map Consider the precollisional states M 1 M. Return time function: R : M 1 M 1 such that R(x) = n if the lower ball hits the floor n + 1 times between the next collision and the one after that. Partition of M 1 : M 1 = n=0 R n where R n = {x M 1 R(x) = n}. First return dynamics: ˆT : M 1 M 1 such that ˆT (x) = (F R(x) 2 F 1 )(x).

10 Figure of the phase space

11 The way of the proof seven conditions such as: Growth Lemma like condition & uniform hyperbolicity in T Chernov & Zhang Existence of 0 1 with exponential tail bound Young I. T enjoys exponential decay polynomial tail bound for 0 Young II. polynomial mixing rate for T polynomial tail bound for the reaching times of 1 by T

12 The magical condition : Growth lemma ˆT is uniformly hyperbolic.

13 The magical condition : Growth lemma ˆT is uniformly hyperbolic. ˆT expands unstable manifolds, but meanwhile the singularities of it may cut them into pieces. To construct a Young-tower one has to show that expansion prevails cuting. Due to Chernov and Zhang it is enough to check the following: lim inf δ 0 sup W : W <δ i Λ 1 i < 1

14 ˆT is not enough There are two cases: 1. short unstable manifolds close to the accumulation point of singularities

15 ˆT is not enough There are two cases: 1. short unstable manifolds close to the accumulation point of singularities are done because Λ Rn n 2.

16 ˆT is not enough There are two cases: 1. short unstable manifolds close to the accumulation point of singularities are done because Λ Rn n but ˆT does not satisfy the inequality on R 0 R

17 Second iterate

18 Estimates

19 Köszönöm a figyelmet!

20 Köszönöm a figyelmet! in English: Thank you for your attention!