Introduction
Setup
The result
Induced map
Proof
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
11.08.2011.
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
The system of two falling balls
v2
"+"
m2
h2
gravity
v1
m1
h1
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Brief history
I
For n balls (instead of just two) if m1 > m2 > · · · > mn then
the system is hyperbolic (Wojtkowski 1990).
I
It is enough to assume that m1 ≥ m2 ≥ · · · ≥ mn , but at least
one inequality is strict (Simányi 1996).
I
For m1 = m2 the system is integrable, and if m1 < m2 then
elliptic periodic orbit exists.
I
The system of two falling balls is ergodic iff m1 > m2
(Liverani-Wojtkowsi 1995).
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Poincaré-section and Wojtkowski coordinates
Let m := m1 and set m1 + m2 = 1.
Also set J := mgh1 + 12 mv12 + (1 − m)gh2 + 12 (1 − m)v22 = 21 .
The phase space M̃ = (h1 , v1 , h2 , v2 )|0 < h1 < h2 , J = 12 .
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Poincaré-section and Wojtkowski coordinates
Let m := m1 and set m1 + m2 = 1.
Also set J := mgh1 + 12 mv12 + (1 − m)gh2 + 12 (1 − m)v22 = 21 .
The phase space M̃ = (h1 , v1 , h2 , v2 )|0 < h1 < h2 , J = 12 .
A Poincaré-section M := (h1 , v1 , h2 , v2 ) ∈ M̃|h1 = 0, v1 > 0 .
In Wojtkowski coordinates h = 12 mv12 and z = v2 − v1 we have
V
M = (h, z)|(0 < h < J) (J − h > 21 (1 − m)v22 ) .
These coordinates are invariant between collisions.
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Expressing the dynamics
Balls won’t collide
M2 ⊂ M
DF2 =
1
q
2
− hm
!
0
1
Balls will collide
M1 ⊂ M

−1
√
DF1 =  q 2
m
h
1− m
+αz 2
2mαz
√
−1 −


q 2 2αz
h
2
1− m +αz
det(DFi ) = 1 hence Lebesgue-measure is invariant.
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
The phase space
m = 0.75
z
2
1
0.1
0.2
0.3
0.4
0.5
0.6
h
-1
-2
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Result
Theorem
There is an open interval I ⊆ ( 12 , 1) such that for all m ∈ I the
system enjoys polynomial decay of correlations
Z
Z
Z
3
f · (g ◦ T n )d µ − fd µ · gd µ ≤ Cf ,g · log (n)
n2
M
M
M
and satisfies the central limit theorem, for Hölder observables.
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
First return map
Consider the precollisional states M1 ⊂ M.
Return time function: R : M1 → M1 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 M1 : M1 = ∪∞
n=0 Rn where Rn = {x ∈ M1 |R(x) = n}.
First return dynamics: T̂ : M1 → M1 such that
R(x)
T̂ (x) = (F2
◦ F1 )(x).
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Figure of the phase space
0.0
M2
-0.5
R0
-1.0
R1
-1.5
R2
-2.0
0.0
0.1
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
0.2
0.3
0.4
0.5
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
The way of the proof
seven conditions such as:
Growth Lemma -like condition &
`
uniform hyperbolicity Hin T L
Chernov & Zhang
Existence of D0 ÍM1
with exponential tail bound
Young I.
`
T enjoys exponential decay
polynomial
tail bound
for D0 ÍM
Young II.
polynomial
mixing rate
for T
polynomial tail bound for
the reaching times of M1 by T
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
The magical condition : Growth lemma
T̂ is uniformly hyperbolic.
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
The magical condition : Growth lemma
T̂ is uniformly hyperbolic.
T̂ 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:
X
lim inf sup
Λ−1
<1
i
δ→0 W :|W |<δ
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
i
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
T̂ is not enough
There are two cases:
1. short unstable manifolds close to the accumulation point of
singularities
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
T̂ is not enough
There are two cases:
1. short unstable manifolds close to the accumulation point of
singularities are done because Λ|Rn n2 .
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
T̂ is not enough
There are two cases:
1. short unstable manifolds close to the accumulation point of
singularities are done because Λ|Rn n2 .
2. but T̂ does not satisfy the inequality on ∂R0 ∩ ∂R1 .
2.0
1.5
1.0
0.5
0.6
0.7
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
0.8
0.9
1.0
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Second iterate
0.1
0.2
0.3
0.4
0.5
-0.5
-1.0
-1.5
-2.0
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Estimates
1.4
1.3
1.2
1.1
1.0
0.9
0.6
0.7
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
0.8
0.9
1.0
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Köszönöm a figyelmet!
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics
Introduction
Setup
The result
Induced map
Proof
Köszönöm a figyelmet!
in English:
Thank you for your attention!
Péter Bálint, Gábor Borbély, A. N.V.
Statistical properties of the system of two falling balls
Budapest University of Technology and Economics