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Overview of the Random Coupling Model. Jen-Hao Yeh, Sameer Hemmady, Xing Zheng , James Hart, Edward Ott, Thomas Antonsen, Steven M. Anlage. Research funded by AFOSR and the ONR/UMD AppEl, ONR-MURI and DURIP programs. “ray chaos”. In the ray-limit it is possible to define chaos. - PowerPoint PPT Presentation
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Overview of the Random Coupling Model
Jen-Hao Yeh, Sameer Hemmady, Xing Zheng, James Hart, Edward Ott, Thomas Antonsen, Steven M. Anlage
Research funded by AFOSR and the ONR/UMD AppEl, ONR-MURI and DURIP programs
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It makes no sense to talk about“diverging trajectories” for waves
1) Classical chaotic systems have diverging trajectories
Regular system
2-Dimensional “billiard” tables with hard wall boundaries
Newtonianparticletrajectories
Wave Chaos?
2) Linear wave systems can’t be chaotic
3) However in the semiclassical limit, you can think about raysWave Chaos concerns solutions of wave equations which, in the semiclassical
limit, can be described by chaotic ray trajectories
qi+Dqi, pi +Dpiqi, pi
Chaotic system
qi, pi qi+Dqi, pi +Dpi
In the ray-limitit is possible to define chaos
“ray chaos”
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Ray Chaos
Many enclosed three-dimensional spaces display ray chaos
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• Wave Chaotic Systems are expected to show universal statistical properties, as predicted by Random Matrix Theory (RMT) Bohigas, Giannoni, Schmidt, PRL (1984)
UNIVERSALITY IN WAVE CHAOTIC SYSTEMS
• RMT predicts universal statistical properties:Closed Systems
Open Systems• Eigenvalue nearest
neighbor spacing• Eigenvalue long-range
correlations• Eigenfunction 1-pt, 2-pt
correlations• etc.
• Scattering matrix statistics: |S|, fS
• Impedance matrix (Z) statistics (K matrix)
• Transmission matrix (T = SS†), conductance statistics
• etc.
H
The RMT Approach: Wigner; Dyson; Mehta; Bohigas …
Complicated Hamiltonian: e.g. Nucleus: Solve
Replace with a Hamiltonian with matrix elements chosen randomlyfrom a Gaussian distribution
Examine the statistical properties of the resulting Hamiltonians
EH
0.5 1.0
0.5
1.0
0
0.5000
1.0001000
500
0
12MLE
72.0)/( 22 D Qkk n
0.5 1.0
0.5
1.0
0
0.5000
1.0001000
500
0
12MLE
72.0)/( 22 D Qkk n
12MLE
72.0)/( 22 D Qkk n
2T
1T
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Billiard
IncomingChannel
OutgoingChannel
Chaos and ScatteringHypothesis: Random Matrix Theory quantitatively describes the statistical
properties of all wave chaotic systems (closed and open)
|S|S1111|||S|S2222||
|S|S2121||
Frequency (GHz)
|| xxS
|S|S1111|||S|S2222||
|S|S2121||
Frequency (GHz)
|| xxS
Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency
B (T)
Transport in 2D quantum dots: Universal Conductance Fluctuations
Res
ista
nce
(kW
) mm
S matrix
NN V
V
V
S
V
V
V
2
1
2
1
][
S matrix
NN V
V
V
S
V
V
V
2
1
2
1
][
Incoming Voltage waves
Outgoing Voltage waves
Nuclear scattering: Ericson fluctuations
dd
Proton energy
Compound nuclear reaction
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Incoming Channel
Outgoing Channel
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The Most Common Non-Universal Effects:1) Non-Ideal Coupling between external scattering states and internal modes (i.e. Antenna properties)
Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details
We have developed a new way to remove these non-universal effects using the Impedance Z
We measure the non-universal details in separate experiments and use them to normalize the raw impedanceto get an impedance z that displays universal fluctuating properties described by Random Matrix Theory
Port
Ray-Chaotic Cavity
Incoming wave
“Prompt” Reflection due to
Z-Mismatch between antenna
and cavity
Z-mismatch at interface of port and cavity.
Transmitted wave
ShortOrbits
2) Short-Orbits between the antenna and fixed walls of the billiards
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N-Port Description of an Arbitrary Scattering System
N – Port
System
N Ports Voltages and Currents, Incoming and Outgoing Waves
Z matrix
NN I
II
V
VV
2
1
2
1
][
S matrix
NN V
V
V
S
V
V
V
2
1
2
1
][
1V
1VV1 , I1
VN , INNVNV
)()( 01
0 ZZZZS
)(),( SZ Complicated
Functions of frequency
Detail Specific (Non-Universal)
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Step 1: Remove the Non-Universal CouplingForm the Normalized Impedance (z)Coupling is normalized away at all energies
Port
ZCavity
Port
ZRad
CavityCavityCavity XjRZ
RadRadRad XjRZ
The waves donot return to the port
RadiationLosses
ReactiveImpedance ofAntenna
Perfectly absorbingboundary
Cavity
Rad
RadCavity
Rad
Cavity
RXX
jRR
z
Combine
X. Zheng, et al. Electromagnetics (2006)
ZRad: A separate, deterministic, measurement of port properties
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Prob
abili
ty D
ensi
ty
-2 -1 0 1 20.0
0.3
0.62a=0.635mm
2a=1.27mm
)Im(z-500 -250 0 250 500
0.000
0.005
0.010
0.015
2a=1.27mm
2a=0.635mm
))(Im( WCavZ
Testing Insensitivity to System Details
CAVITY BASE
CrossSection View
CAVITY LID
Radius (a)
CoaxialCable Freq. Range : 9 to 9.75 GHz
Cavity Height : h= 7.87mm Statistics drawn from 100,125
pts.
Rad
RadCavity
Rad
Cavity
RXX
jRR
z
RAW Impedance PDF NORMALIZED Impedance PDF
Metallic Perturbations
Port 1
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Step 2: Short-Orbit TheoryLoss-Less case (J. Hart et al., Phys. Rev. E 80, 041109 (2009))
Original Random Coupling Model:
where is Lorentzian distributed (loss-less case)
vTTviXZ Radpavg
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Now, including short-orbits, this becomes:
RadRvv
RadRadRad RRiiXZ 0
0
pavgpavgpavg RRiiXZ
where is a Lorentzian distributed random matrix projected into the2L/l - dimensional ‘semi-classical’ subspace
with
1
2
vTvZRad is the ensemble average of the semiclassical Bogomolny transfer operatorT
… and can be calculated semiclassically…TExperiments: J. H. Yeh, et al., Phys. Rev. E 81, 025201(R) (2010); Phys. Rev. E 82, 041114 (2010).
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Applications of Wave Chaos Ideas to Practical Problems
1) Understanding and mitigating HPM Effects in electronicsRandom Coupling Model
“Terp RCM Solver” predicts PDF of induced voltagesfor electronics inside complicated enclosures
2) Using universal statistics + short orbits to understand time-domain dataExtended Random Coupling Model
Fading statistics predictionsIdentification of short-orbit communication paths
3) Quantum graphs applied to Electromagnetic Topology
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Conclusions
Demonstrated the advantage of impedance (reaction matrix)in removing non-universal features
The microwave analog experiments can provide clean, definitivetests of many theories of quantum chaotic scattering
Some Relevant Publications:S. Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005)
S. Hemmady, et al., Phys. Rev. E 71, 056215 (2005)Xing Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 3 (2006)Xing Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 37 (2006)
Xing Zheng, et al., Phys. Rev. E 73 , 046208 (2006)S. Hemmady, et al., Phys. Rev. B 74, 195326 (2006)
S. M. Anlage, et al., Acta Physica Polonica A 112, 569 (2007)
Many thanks to: R. Prange, S. Fishman, Y. Fyodorov, D. Savin, P. Brouwer, P. Mello, F. Schafer,
J. Rodgers, A. Richter, M. Fink, L. Sirko, J.-P. Parmantier
http://www.cnam.umd.edu/anlage/AnlageQChaos.htm
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The Maryland Wave Chaos Group
Tom Antonsen Steve AnlageEd Ott
Elliott Bradshaw Jen-Hao Yeh James Hart Biniyam Taddese