PHYSICAL REVIEW E 75, 046204 共2007兲

Leaking billiards
Jan Nagler, Moritz Krieger, Marco Linke, Johannes Schönke, and Jan Wiersig
Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee, D-28334 Bremen, Germany
共Received 13 December 2006; published 10 April 2007兲
Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid
walls. However, billiard walls are not always completely reflective and measurements inside can also open thebilliard. Since boundary openings have been studied extensively in the literature, we rather model leakages
inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family ofbilliard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard
can be tuned from stadium 共being fully hyperbolic兲 over circle 共integrable兲 to the lemon-shaped billiard 共mixed
chaotic兲. For the stadium billiard we found an algebraically decaying mean escape time with the linear size ⑀of the leakage 具nesc典 ⬃ ⑀−1 together with an exponential decay of the survival probability distribution. The
finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller
than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due tothe mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of
the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean
escape time, 具nesc典 ⬃ 1 − ⑀, which we refer to as flooding law. Our findings are helpful in understanding
dynamics of leaking Hamiltonian systems.
DOI: 10.1103/PhysRevE.75.046204

PACS number共s兲: 05.45.Ac, 05.45.Pq

I. INTRODUCTION

where chaoticity is desired to significantly enhance the emission directionality 关22兴. Stadium-shaped microcavities havebeen studied in a number of publications 关23兴. For several
geometries the dynamics of billiards with a single hole in theboundary have been investigated in the context of the decayof chaotic systems and chaotic wave scattering 关24–26兴. In
order to demonstrate the existence of Maxwell’s daemon inchaotic billiards, Zaslavsky coupled two billiards through asmall hole 关27兴. Interestingly, the daemon accomplishes a
difference in the pressures of the two chaotic particle gases.
A beautiful example for a two-hole leaking of the circle billiard represents Bunimovich and Dettmann’s demonstrationof a direct connection between leaking billiard experiments
and the Riemann hypothesis 关28兴. We also mention works on
chaotic scattering in ballistic circle and stadium microstructures, cross junctions, and cut-circle billiards 关29–33兴. Electric density measurements of leaking microwave billiards inconfiguration space have been performed by Alt and coworkers 关34,35兴.
Besides billiards, leaking Hamiltonian systems have beenstudied in different contexts, e.g., for the standard map, therestricted three-body problem of celestial mechanics, andtransport problems in incompressible fluids 关36,37兴. It has
been shown that suitable Poincaré surface of sections where

1539-3755/2007/75共4兲/046204共7兲

stadium

circle

lemon

s

2a
s=1/2

Billiard systems have been intensively studied over several decades both experimentally and theoretically. In thegreat majority of billiards considered in the literature a perfectly reflecting boundary encloses a domain in which thelong-term dynamics of a freely moving particle or wave isunder investigation. Examples for integrable 共regular兲 billiards represent rectangular, elliptic, and circular geometries.
In almost every case, the slightest deformation makes themnonintegrable. Consider, for example, the stadium billiard in
Fig. 1. Whereas every starting position in phase-space of thecircle billiard 共a = 0兲 corresponds to regular motion, for a⬎ 0 the motion is ergodic, mixing, and exponentially unstable with Lyapunov exponent ␭ ⬃ a1/2 for a Ⰶ 1 关1兴.
Theory and experiments on the wave dynamics in microwave cavities and quantum dots have greatly benefited fromeach other with the consequence of a considerable number ofpublications 关2–7兴, see Ref. 关8兴 for an overview. Also, opticalbilliard experiments have attracted enormous interest. In particular, we refer to Refs. 关9,10兴 where ultracold atoms
bounce off beams of light that form the boundary of thebilliard. Experiments performed by Chinnery and Humphrey
are worth noting since they provide a mesoscopic approachto the field that allows for excellent comparison between
experiment or numerics and theory. They investigated acoustic resonances of an insonified water-filled stadium billiard
with a schlieren visualizing system 关11兴. Moreover, much
effort has been devoted to classical, semiclassical, and quantum mechanical studies on regularity and chaos in billiards
关12–21兴.
Billiards whose walls are not fully reflective, or completely closed, play an important role as models to study
chaotic phenomena in open systems but also have a vast bulkof applications, e.g., in the field of optical microresonators

R

2(a+R)

s=0

R

α

a>0

a=0

a<0

FIG. 1. 共Color online兲 Schematic representation of stadium 共a⬎ 0兲, circle 共a = 0兲, and lemon-shaped 共a ⬍ 0兲 billiard.

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NAGLER et al.

points are colored with respect to the corresponding orbit’s
escape time reveal computationally quickly a backbone of
the system’s phase-space and provide considerable insight in
the leaking dynamics of the system. In the next sections weapply this approach to the stadium-lemon-billiard family, seeFig. 1. Instead of opening the boundary we introduce a small
leakage inside the billiard. An experimental realization could
be a microwave billiard where the antenna plays the role ofthe internal leakage 关34,35兴.
II. MODEL

Bunimovich first considered a stadium billiard where twosemicircular arcs with radii R are joined by tangential
straight lines with lengths 2a ⬎ 0 关38兴, see Fig. 1. The limiting case a = 0 is the integrable circle billiard. Recent interest
has been devoted to the case −2 ⬍ a ⬍ 0 where straight lines
no longer exist but two circular sectors form a boundary
which resembles a lense or lemon 关14,17兴.
The phase-space of this one-parameter family of billiards
is conveniently described by Birkhoff’s coordinates: the共normalized兲 arc-length coordinate 0 艋 s ⬍ 1 along theboundary and p ⬅ cos共␣兲 being the momentum coordinate
共energy is scaled out兲 where 0 ⬍ ␣ ⬍ ␲ is the incident angle.
With this choice of coordinates the phase-space mapping
M : 共sn , pn兲 → 共sn+1 , pn+1兲 is symplectic 关12兴. The radii R in
Fig. 1 can be scaled out and the cases p = ± 1 correspond toorbits being tangential to the boundary. Hence throughout the
paper we assume unit system radii R ⬅ 1 and consider thephase-space specified by 0 艋 s ⬍ 1, and −1 ⬍ p ⬍ 1.
For the nontrivial cases a ⫽ 0, there are two reflexion
symmetries that can be expressed by the operation
S : 共s , p兲 → f共s , p兲 where

f共s,p兲 =冦

共s,p兲

for 0 艋 s ⬍ 1/4,

共1/2 − s,− p兲 for 1/4 艋 s ⬍ 1/2,
共s − 1/2,p兲

for 1/2 艋 s ⬍ 3/4,

共1 − s,− p兲

for 3/4 艋 s ⬍ 1.

冧

1

3

4

5

1

2

3

4

5

FIG. 2. 共Color online兲 Schematic view of the leakage positions
共disks兲. Half of the system size is portioned in such a way that fivedisks are placed equidistantly, disk 1 is centered, and disk 5 istangent to the boundary.
A. Stadium billiard

Figure 4 displays the ETDs for the stadium billiards. The
white horizontal stripes in the ETDs represent the 共marginally stable兲 orbits created by perpendicular reflections between the stadiums’ straight line segments. These orbits arecalled bouncing-ball orbits. The dimension of the leakage isimprinted in the diagrams: The width of the slits in the upper
ETDs represents the hole diameter 2⑀. The stripe-shaped regions in black represent particles that leak out after a single
reflection. The upper diagram line corresponds to billiards
that are leaked by circular middle holes of radii ⑀ = 0.01,
⑀ = 0.05, ⑀ = 0.1, and ⑀ = 0.2. If the circular leakages are replaced by quadratic leakages of equal areas, corresponding
共magnifications of the兲 ETDs are virtually unaffected 共notshown兲. As illustrated in the lower diagram line in Fig. 4 theposition of the leakage 共see Fig. 2兲 has a widely more profound impact on the diagram structure than the leakage geometry. Note that shifting the hole away from the center ofthe billiard breaks one of the reflexion symmetries.
Figure 5 shows the dependence of the escape time diagrams on the control parameter a for the stadium casea ⬎ 0. Though the phase-space is known to be hyperbolic forany a ⬎ 0, the escape time diagrams demonstrate the exis-

共1兲

+1

Note that due to the arclength normalization, S is independent of geometry parameter a. In addition to the reflexion
symmetries, the billiard exhibits a time-reversal symmetry
T : 共s , p兲 → 共s , −p兲. It is important to mention that, first,
throughout the paper we partly break the symmetries byplacing a leakage inside the billiard, and second, the leakage
can be formally considered an internal boundary.

p

1

0

−1

0
0

III. ESCAPE TIME DIAGRAMS

We now investigate position and extent of escape orbits
for the case that a circular leakage with radius ⑀ is placed in
the configuration space. We consider different positions of
the circular leakage as illustrated in Fig. 2.
Figure 3 shows an escape time diagram 共ETD兲 for the
共s , p兲 plane. Each point represents a starting point in phasespace. The color codes the number of reflections before escape.

2

s

1

FIG. 3. 共Color online兲 Ranges and color legend for escape time
diagrams 共ETD兲. Parameter values for the displayed example withcentral circular hole: a = −0.2, hole radius ⑀ = 0.05, and Nmax
= 10 000. The color legend displays the normalized escape timenesc / Nmax where nesc is the number of reflections before escape, andNmax denotes the maximal number of reflections during the simulation. The lighter the color the longer the particle is trapped in thebilliard. White colored phase-space points correspond to nesc
= Nmax, black ones correspond to nesc = 1.

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LEAKING BILLIARDS
a=−1

a=−0.5

2ε

2

a=0

3
1

4

a=0.5

FIG. 4. 共Color online兲 Escape time diagrams for the Birkhoff
phase space, i.e., the 共s , p兲 plane, and for a = 1. The color of a point
codes the number of reflections before escape through a circular
central leakage 共upper diagram line兲, and decentrally placed leakages 共lower line兲. White points represent bounded motion 共maximal
number of specular reflection is Nmax = 10 000兲, all other points represent starting positions of escape orbits. The lighter the color, thelarger the number of reflections 共see scale in Fig. 3兲: black
共nesc = 1兲, white 共nesc = Nmax兲. Upper line, from left to right: holeradius ⑀ = 0.01, ⑀ = 0.05, ⑀ = 0.1, and ⑀ = 0.2. Lower line: decentral
disk positions 2, 3, 4, and 5 共see Fig. 2兲 for fixed radius ⑀ = 0.1.

tence of a rather complex topology on position and extent oflong- and short-lived motion. Figure 5 displays the situationfor eight different values of a. First, the regular phase-space
for a = 0 共see black stripe in Fig. 6兲 is more and more raggedwith increasing parameter value a. Second, the dominance oflong-lived motion of the circularlike geometries a ⬇ 0 is destroyed by the onset of escape regions. The inset in the ETD
for a = 0.1 in Fig. 5 displays the ETD when the leakage isplaced tangential to the boundary 共see Fig. 2, position 5兲—rather than central. As a consequence, even for values of aclose to zero, regions of long-lived motion are displaced byregions of escape due to tangential leakage of the so-called
whispering-gallery orbits with p ⬇ ± 1.
B. Lemon billiard

We now investigate the mixed phase-space domain of thestadium-lemon-billiard family, that is, the parameter region

pos5

FIG. 5. 共Color online兲 Escape time diagrams for the stadiumbilliard with central circular hole of radius ⑀ = 0.05 for varying values of a ⬎ 0. Values for a from left to right: 0.1, 0.2, 0.3, 0.4 共upper
diagram line兲, 0.5, 0.8, 1.4, 2.1 共lower diagram line兲. Inset: ETD for
a = 0.1 but for the tangential leakage position 5.

a=1

FIG. 6. 共Color online兲 Iteration maps for lemon-shaped
共a = −1.0, a = −0.5兲, circle 共a = 0兲, and stadium 共a = 0.5, a = 1.0兲billiard for a centered 共position 1 in Fig. 2兲 circular leakage withradius ⑀ = 0.05. The diagram plane is the 共full兲 phase-space region
共0 ⬍ s 艋 1, −1 ⬍ p ⬍ 1兲 共see also Fig. 3兲. Black points indicate anescape after a single reflection when the trajectory begins at thecorresponding starting position in phase-space 共s , p兲. Red points
represent trajectories which end up in the hole after two reflections,green and blue correspond to three and four reflections, respectively. White indicates a survival after four reflections.

a ⬍ 0. In Fig. 7 we provide a phase-space overview displaying ETDs for the range −2 ⬍ a ⬍ 0. The phase-space structureof each diagram is partitioned into three main domains.
White colored island chains of regular 共almost quasiperiodic兲
motion with the corresponding periodic centers 共not shown兲
in a sea of highly structured chaos 共light or blue regions兲.
The linear scale of the leakage is represented in the plain
unstructured 共nonfractal兲 escape basins in black or dark blue.
The insets in the ETD for a = −0.1 in Fig. 7 show the behavior for a → −0. For examples of corresponding orbits 共that
are not hindered by the leakage兲 we refer to the bulk ofliterature triggered by the coining paper, Ref. 关14兴.
We now investigate how the position of the leakage affects the phase-space structure in the ETDs for a ⬍ 0. In Fig.
8 we show for five different values of the geometry parameter a four diagrams corresponding to four off-centered leakage positions. More precisely, each diagram line provides the
ETDs for the leakage positions 2, 3, 4, and 5 as explained inFig. 2.
Under the shift of the leakage from the center to the border, islands representing never escaping orbits 共partly兲 disappear when their corresponding quasiperiodic orbits are
共partly兲 blocked by the leakage in configuration space, see
the lower diagram line 共e兲 in Fig. 8. In contrast to that behavior, lemon billiards with phase-space that is dominated bychaotic motion are not profoundly affected by the leakage
position, see diagram line 共b兲 corresponding to a = −1 in Fig.
8.
IV. MEAN ESCAPE TIME DISTRIBUTIONS

For billiards the 共mean兲 escape time distribution has beenstudied both experimentally and theoretically, see in addition
to the previously cited works, Ref. 关39兴 and references
therein. To be precise, we define the mean escape time as

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NAGLER et al.
a)

a)

b)

b)

c)
c)

d)
d)

e)

a=−0.05a=−0.03a=−0.01

FIG. 7. 共Color online兲 Escape time diagrams for various parameter values a ⬍ 0 for fixed circular hole radius, ⑀ = 0.05, that is centered. Nmax = 10 000. Left to right: −1.97, −1.9, −1.8, −1.7 共a兲, −1.6,−1.5, −1.4, −1.3 共b兲, −1.2, −1.1, −1.0, −0.9 共c兲, −0.8, −0.7, −0.6,−0.5 共d兲, −0.4, −0.3, −0.2, −0.1 共e兲. Insets: ETD for a = −0.05,−0.03, and a = −0.01.

具nesc典 ⬅ 具nesc共s,p兲典/Nmax ,共2兲

where brackets 具·典 on the right hand side denote the averageover the phase-space 共0 艋 s ⬍ 1, −1 ⬍ p ⬍ 1兲 and nesc共s , p兲 isthe escape time 共number of reflections兲 for each starting
phase-space position 共s , p兲 of the trajectory. As throughout
the paper, Nmax is the maximal number of reflections.

e)FIG. 8. 共Color online兲 Escape time diagrams for four decentral
positions of a circular hole. ⑀ = 0.05 is fixed, N = 10 000. Left to rightin a diagram line: leakage position 2, 3, 4, and 5. a = −1.9 共a兲,
a = −1.0 共b兲, a = −0.7 共c兲, a = −0.5 共d兲, a = −0.2 共e兲.

denote the probability that the particle leaves the system aftern elastic reflections. Denoting with p the probability to leave
the system in a time step, the survival probability is thengiven bynⰇ1

Psurv共n兲 = 共1 − p兲n ⯝ e−pn .

共3兲

The probability for the particle not to leave the systemthrough the leakage during the first n − 1 reflections, but at

A. Phase-space dominated by chaotic motion1
a=1 pos 1
pos 2
pos 3
pos 4
pos 5
a=0.5 pos 1
pos 2
pos 3
pos 4
pos 5
1/ε

0.8

Mean escape time

In Fig. 9 we plot the mean escape time vs the leakage sizefor five different hole positions. As expected, for the hyperbolic cases a = 0.5 and a = 1 all curves are close to a powerlaw behavior. The power-law approximation of the decay isbetter the smaller the hole size: in Fig. 10 we plot the scaled
mean escape time for a = 1 vs the hole radius for small valuesof the leakage size 10−4 艋 ⑀ ⬍ 10−1. As can be seen in the
log-log-plot, for small leakage sizes, the mean escape time
decays as a power of −1. For large hole sizes, that is, of theorder of the system size, finite-size effects cause deviations
from the theoretical curve.
In the following we show that it is sufficient to consider
the billiard map with strongly chaotic dynamics as a stochastic map in order to explain the power law for the mean escape time dependence on the leakage dimension. Let P共n兲

0.6

0.4

0.2

0
0

0.1

0.2

0.3

Hole size ε

0.4

0.5

FIG. 9. 共Color online兲 Mean escape time distribution dependence on circular leakage position for a = 0.5 共solid symbols兲, andfor a = 1 共open symbols兲. The bold line is the power law 1 / ⑀.
Curves are arbitrarily normalized to coincide at 具nesc典 = 1.

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LEAKING BILLIARDS-1Mean escape time

10

data
power law 1/ε

-2

10

-3

10

-4

10

-4

-3

10

10

-2

Hole size ε

-1

10

10

FIG. 10. 共Color online兲 Mean escape time distribution for small
circular leakages in a double-logarithmic plot 共a = 1.0兲. For small
leakage sizes data are in good agreement with the theoretical curve
共power law兲.

time step n, is given by P共n兲 = Psurv共n − 1兲 ⫻ p. Putting this
together we have⬁具nesc典 = 兺 nP共n兲
n=1

⬁

p
=
兺 n共1 − p兲n1 − p n=1
=
=

冋

⬁

d
p
q 兺 qn1 − p dq n=0

冋

册

册

q=1−p

q
p
1 − p 共1 − q兲2 q=1−p

1
= ,
p

共4兲

where we here consider the limit Nmax → ⬁. Since the hole
represents a stripe in phase-space, the escape probability ineach time step is p = ⑀ / A, A being the total area in phasespace. Hence we obtain 具nesc典 ⬃ ⑀−1 in good agreement withthe data for ⑀ Ⰶ 1.

FIG. 11. 共Color online兲 Mean escape time distribution for
a = 0.1 and five leakage positions. The bold lines are the theoreticalcurves 1 − ⑀ and 1 / ⑀, respectively. Curves are arbitrarily normalized
to coincide at 具nesc典 = 1.

motion have distributions that follow the 1 − ⑀ law, and systems with phase-space dominated by chaotic motion have
distributions that can be well approximated by the 1 / ⑀ law.
Note that in contrast to the survival probability function
Psurv共t兲 measurements in the literature for different mixed
chaotic systems 关39兴, P共⑀兲 cannot be approximated by simple
crossovers of power laws or exponential decays. The finestructure of the P共⑀兲 curves for the mixed phase-space mirrors the dependence on position and extent of resonances,and leakage position.
If the phase-space is dominated by regular regions, leaking the phase-space has the effect that the regular regions in
the Poincaré surface of section are flooded by regions ofescape. For the integrable circle billiard, being the casea = 0, this fact is straightforward to see. Consider a hole in
the unit circle billiard centered in the middle, with radius⑀ ⬍ 1. Trajectories that have a tangential intersection with thehole must satisfy
兩cos共␣兲兩 = ⑀ ,共5兲

if ␣ denotes the incident angle, see Fig. 1. Due to the identityp ⬅ cos共␣兲, for every starting position s on the unit circle,

B. Phase-space dominated by regular motion1

0.8

Mean escape time

Figure 11 shows the mean escape distribution for the quasiregular case a = 0.1. We mean by quasiregular the casewhen the Lyapunov exponent of a hyperbolic system is positive but close to zero 共as for a → 0兲. As a consequence, theETDs are dominated by quasiregular orbits 共see, e.g., light or
blue regions in Fig. 5兲. In contrast to Fig. 9, only for thetangential leakage position can the mean escape distribution
be approximated by the 1 / ⑀ power law. We can understandthis behavior because for the billiard with the tangential leakage whispering-gallery orbits can leave the system 共see also
inset in Fig. 5兲. All other data curves corresponding to positions 1–4 共see Fig. 2兲 lie between the 1 / ⑀ law and a 1 − ⑀curve, the origin of which will be explained later.
In Fig. 12 we plot the mean escape times of billiard systems whose distributions can be well approximated by either
theoretical curve. Strikingly, systems dominated by regulara = 0.0
a = -0.1
a = -0.2
a = 0.5
a = 0.7
a = 0.9
a = 1.1
1-ε, 1/ε

0.6

0.4

0.2

0
0

0.1

0.2

0.3

Hole size ε

0.4

0.5

FIG. 12. 共Color online兲 Mean escape time distribution. Average
escape time vs linear hole size. The bold lines are the theoreticalcurves 1 − ⑀ and 1 / ⑀, respectively. Curves are arbitrarily normalized
to coincide at 具nesc典 = 1.

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trajectories with momentum −⑀ 艋 p 艋 ⑀ will escape throughthe leakage. As a consequence, the fraction of escape orbits
Aesc in the 共s , p兲 plane, 0 艋 s ⬍ 1, −1 ⬍ p ⬍ 1, is Aesc = 2⫻2 ⑀ = ⑀from which we read off the mean escape time 共in the limit
Nmax → ⬁兲 as 具nesc典 = 1 − ⑀.
V. CONCLUSIONS

We studied the dynamics of leaking billiards. Rather thanopen the boundary we considered leakages inside the billiard. Our investigation was motivated by works on leaking
Hamiltonian systems together with recent applications. As asuitable one-parameter billiard system we focused on thestadium-lemon-billiard family as a prominent chaotic systemthat can be tuned from fully hyperbolic behavior to a mixed
chaotic behavior. In the context of leaking Hamiltonian systems 关37兴, it is known that leaking is capable to reveal thephase-space structure 共i.e., the chaotic saddle兲, here represented in terms of escape time diagrams. We demonstrated
how the position and geometry of the leakage affects the

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dynamics. The mean escape time dependence on the size ofthe leakage is algebraic—as known for openings of theboundary for hyperbolic systems 关24兴. This finding corresponds to an exponentially decaying survival probability distribution and could be fully explained by a stochastic map
approach. On the other hand, for systems where the phasespace consists to a large extent of regular orbits, we numerically found a linear dependence of the mean escape time on
the leakage dimension as 具nesc典 ⬃ 1 − ⑀. Regions of regularorbits in phase-space are flooded by escape domains as theleakage increases. For the integrable circle billiard we comprehended the phenomenon analytically.
In a slightly broader context, our investigations extend
recent work on time continuous leaking systems 关36兴 andrepresent examples for leaking Hamiltonian systems in theirconfiguration space.
ACKNOWLEDGMENT

We gratefully acknowledge Peter H. Richter for helpfulcomments and for the careful reading of the manuscript.

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