Long Time Survival Probability for the Stadium Billiard

1
Long Time Survival Probability for the Stadium
Billiard

• By Orestis Georgiou
• Supervised by Carl Dettmann
• arXiv0812.3095
• Subjects Chaotic Dynamics (nlin.CD)

Mathematics and Physics of Billiard-like systems
09 Universidade Estadual Paulista Programa de
Pós-Graduação em Física
2
Open Billiards

• Systems with holes or leakages introduced in
  1980s
• Applications Statistical Mechanics, Atom Optics,
  Laser Optics, Acoustics, Quantum Dots
• Distributions
• Open systems provide a pathway towards
  understanding Chaos and much more.

• escape probability
• survival probability of orbits

"QD-LED"
Electron densities of eight-electron quantum dots
with different geometries
3
History of the Stadium Billiard

• 1974 (L. Bunimovich) Introduced
• 1978 (G. Benettin and J. M. Strelcyn) Entropy
  was numerically estimated
• 1979 (L. Bunimovich) Ergodic properties
  established
• 1979 (S. McDonald) RM conjecture validated on
  this billiard
• 1983 (F. Vivaldi et al.) Origin of Long-Time
  Tails of correlations
• 1984 (E. Heller) Scarred wave-functions where
  first discovered on this billiard
• 1992 (H. Gräff et al.) First microwave
  experiments
• 1994 (U. Smilansky and H. Primack) QM
  investigation of stability of bouncing ball
  orbits
• 1996 (R. Markarian et al.) Chaotic properties of
  the elliptical stadium
• 1997 (G. Tanner) Semi-Classical review of the
  stadium
• 2001 (A. Kaplan et al.) Atom-Optics Stadium
  billiard
• 2001 (C. Miniatura) Wave scarring experiments in
  a stadium-like optical fibre
• 2003 (R. Markarian) Proof of polynomial decay of
  correlations
• 2004 (D. Armstead et al.) Power-law decay
  self-similar distributions in stadium
• 2006 (P. Balint and S. Gouezel) Non-standard
  Limit Theorem for the stadium

Our Contribution We obtain an exact expression
for the leading order term of the long time
survival probability function of the stadium
billiard with a hole on one of its straight
segments.
4
The Set up

• Classical particle of unit mass and unit speed
• Elastic collisions

2r
h1
h2
2a

• the x coordinate takes values from a to a
• small hole in the interval (h1, h2) of size e
• h2h1 e

5
Dynamical portrait
6
The Result

• There are only two families of orbits that define
  the long time Survival Probability
• I. Moving Towards the hole
• II. Moving Away from the hole

7
Numerical Simulations
tP (t)
tP (t)
a 10 r 1 e 2 h1 5
a 10 r 1 e 0.5 h1 -5
tP (t)
a 5 r 1 e 0.5 h1 3
t
t
t
Initial conditions, uniformly distributed along
the billiard boundary. They are followed until
they escape and their individual life-span is
recorded.
8
Moving towards the hole Analytics
Survival probability of the set of orbits
initially approaching the hole from either side.
The orbits will not jump over the hole.
The set of initial conditions (x,?) which will
escape at time t
9
Moving Away from the hole

• The set of initial conditions (x1, ?1),
  initially moving away from the hole, reflecting
  on the semicircular ends of the billiard
• Reflections from the circular arcs cause a
  small change to the initial angles
• The time to escape of an orbit is highly
  sensitive to ?1

10
Moving Away Numerical Simulations
?1
x a
x1
Initial Conditions, that escape without jumping
over the hole
11
Moving Away Numerical Simulations
?1
x1
Fractal pattern appears
12
Moving Away Numerical Simulations
Initial Conditions that survive at least t50
without jumping over the hole
5
4
3
2
1
n0
A series of spikes, each at different angles,
increasing in height with n.
n is the number of non-essential collisions from
straight to straight boundaries.
n 1 n 2 n 3
13
Moving Away from the hole
There are 2 possible scenarios of reflection at
the circular arc.
One collision case f3
Two collision case f4
?4
?1
?3
?1
Note Valid only for small angles. Small angles
remain small.
Koo-Chul Lee (1988)
14
Moving Away from the hole
Time to escape
After small angle approximations
Rearranging this we can obtain two functions ( f3
and f4 ), describing hyperbolas in the x1- ?1
plane.
15
Moving Away from the hole
3
By requiring orbits not to jump over the hole,
the hyperbolas f3 and f4 produce a series of
spikes, each at different angles, increasing in
height with n
2
1
n0
Each spikes boundary is a hyperbola segment
16
Approximations
x
17
Approximating hyperbolas
18
Approximating hyperbolas
The error in our approximations vanishes
asymptotically
19
Current interests
D. N. Armstead, B. R. Hunt and E. Ott
Power-law decay and self-similar distributions in
stadium-type billiards, Physica D Nonlinear
Phenomena V. 193, (2004)
Stadium with a porous vertical wall
Stadium with open-ended sides
For each problem they considered, the asymptotic
similarity of the PDFs, implied the power-law
relaxations and
20
Current interests The open-ended Stadium
Fractal pattern appears distorted
21
A probabilistic model
Armstead et al. suggest a one dimensional map for
the angle parameter and apply it every time the
orbit hits the semi-circular wall
Due to the rapid decay of correlations, we can
consider f, as a uniformly distributed random
variable. and assume that subsequent reflections
are independent of each other.
Future interests
Diffusion in the expanded stadium and
similarities with the I.H.L.G.
22
Comments and Remarks

• The Long time survival probability goes as
• The Constant, depends quadratically on the
  lengths of the straight segments and on the
  position of the hole but not on the holes size.
• The Log3 term is a direct consequence of the
  geometry of the stadium
• The curvature of the boundary near the straight
  segments which lead to a reflected final angle
  which differs by at most a factor of 3.
  
• Any change to that specific section of the
  billiard boundary would change the dynamics
  quantitatively but not qualitatively.
• Expect similar formula for different smooth
  stadium boundaries

Elliptical Stadium
23
Long Time Survival Probability for the Stadium
Billiard
Thank you for your attention
By Orestis Georgiou Supervised by Carl
Dettmann arXiv0812.3095 Subjects (nlin.CD)
Mathematics and Physics of Billiard-like systems
09 Universidade Estadual Paulista Programa de
Pós-Graduação em Física