Geometric Characterization of Nodal Patterns and Domains
description
Transcript of Geometric Characterization of Nodal Patterns and Domains
Geometric Characterization
of Nodal Patterns and
Domains
Y. Elon, S. Gnutzman, C. Joas U. Smilansky
Introduction
• 2002 – Blum, Gnutzman and Smilansky: a “Chaotic billiards can be discerned by counting nodal domains.”
0 jjE
0 Vj
• Research goal – Characterizing billiards by investigating geometrical features of the nodal domains:
• Helmholtz equation on 2d surface (Dirichlet Boundary conditions):
- the total number of nodal domains of . j j
Consider the dimensionless parameter:
jij
ijij E
lA
j
Consider the dimensionless parameter:
jij
ijij E
lA
j
i
• For an energy interval: , define a distribution function:
IEj i
ijjI
Ij
j
NP
1
11
gEEEI ,
• Is there a limiting distribution?
PgEIPE
?
),(,lim
• What can we tell about the distribution?
Rectangle
ynxmNny
Mmx
mn~sin~sinsinsin~
yx
imn
yx
imn
EEmnl
EEmnA
112~1
~12
~~1
)(
2)(
yxmn EEmnE 222 ~~xx
imn ZZmn
mn
1
12~~
~~
2
22)(
EEEEmnmn
yx
imn
1
2~~~~
2
22)(
Rectangle
IEmnI
Imn
mnmn
NP
|~,~
22
~~~~
21
II
dndmdndmmnmn~~~~
2
22
Rectangle
otherwise
PI222
0
84
22
P
Rectangle
222
ddP 1~22
lim0
1. Compact support:
2. Continuous and differentiable
3.
4.
ddP 20
82
lim
Rectangle
mnymnx
imn EEEE
12
)(
• the geometry of the wave function is determined by the energy partition between the two degrees of freedom.
Rectangle
222 ~~ mnEE classmn
quantmn
• can be determined by the classical trajectory alone.
mn
dypm
dxpn
ycl
xcl
21
21
Action-angle variables:
Disc
•
• the nodal lines were estimated using SC method, neglecting terms of order .E1
2)(
)()sin(n
mmn
nmmmn
jE
rjJm
)(
0222
222
)(
0222
222
22
2
1
)tan()21(1)2sin(
84
84
)tan()21(1)2sin(
84
84
8
2)(
C
C
CCdCC
CCdCC
P
n’=1
n’=4
n’=3
n’=2
22)'(
2
'
)'(
arctan1'
)'2sin(11
2
mjmC
C
nm
mn
nmn
Rectangle Disc
Same universal features for the two surfaces:
Disc
222
ddP 1~22
lim0
1. Compact support:
2. Continuous and differentiable
3.
4.
ddP 20
42
lim
n=1m=o
Surfaces of revolution• Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve).
• Same approximations were taken as for the Disc.
mxmnmn sin
xf
Surfaces of revolution• Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve).
• Same approximations were taken as for the Disc.
mxmnmn sin
2
'2
')'(
2 mxfEm
xfE
nmn
nmnnmn
2,122
2222
2,122
2222
2222
2
2
1
8
84
8
84
84
8)(
i z
i z
i
i
dzzG
dzzG
MP
n’=1
n’=2
n’=3
n’=4
• For the Disc:clcl
rclmn EE
rmrE
2
22
21
• For a surface of revolution:
clcl
xclmn EE
xfmxxfE
2
222'1
21
2
2
2rmrE qm
rEErE qmmn
qmr
xfmxE qm2
2
2 xEExE qm
mnqmx
mnmnrnm ~~~~
2
22Rect
Following those notations:
)'2sin(1
12
Disc
Crmn
2
'2
'SOR
2 mxfEm
xfEr
nmn
nmnmn
mnmn ErEErE
21
12
“Classical Calculation”:
0rnm
“Classical Calculation”:
nmT1. Look at
(Classical
Trajectory)
“Classical Calculation”:
0rr
2. Find a point along the trajectory for which:
“Classical Calculation”:
mnmn
r
EE
EE ,
3. Calculate
mnmnr
mn EEEEr
1
20
Separable surfaces
rmn2. can be deduced (in the SC limit)
knowing the classical trajectory solely.
1. In the SC limit, has a smooth limiting distribution with the universal characteristics: - Same compact support:- diverge like at the lower support- go to finite positive value at the upper support
IP
Random waves
rJmbmar m
N
mmm
0sincos
Random waves
Two properties of the Nodal Domains were investigated:
1.Geometrical:
2. Topological: genus – or: how many holes?
P
G=0 G=
2
G=1
Random waves
22
2
Random waves
Random waves
Random waves
Random waves
2
)1(0j
Random waves
Random waves
2
)1(0j
Model: ellipses with equally distributed eccentricity and area in the interval: 19,
2)1(0j
2
212
2
2
min
2
offcut
avg
knl
knA
kd
Random waves
d
Genus
76.4~# c46.4~# c
3.4~# c
262.4~# c
The genus distributes as a power law!
Genus
In order to find a limiting power law – check it on the sphere
Genus
Power law?
Saturation?
?~~# 2gg
AA ~~#Fisher’s exp:
Random waves
1. The distribution function has different features for separable billiards and for random waves.
2. The topological structure (i.e. genus distribution) shows complicate behavior – decays (at most) as a power-low.
P
Open questions:
• Connection between classical Trajectories and .
• Analytic derivation of for random waves.
• Statistical derivation of the genus distribution
• Chaotic billiards.
P