Ultrasound in Regular and Irregular Elastic Bodies Random Matrix Theory and classical elastic waves

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• Ultrasound in Regular and Irregular Elastic
  BodiesRandom Matrix Theory and classical
  elastic waves
• We review laboratory work comparing ultrasonic
  responses in generic elastic bodies to
  predictions from Random Matrix Theory.
  Measurements of level statistics and quantum
  fidelity are found to conform remarkably well to
  RMT even in bodies whose ray dynamics one might
  have considered pseudo integrable or even
  regular. We also find that return
  probabilities, related to the distribution of
  decay rates of excited nuclei, conform well to
  RMT, although a more naive Porter-Thomas-like
  model does almost as well. 

Complex Systems May 2008
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Ultrasound in Regular and Irregular Elastic
BodiesRandom Matrix Theory and classical
elastic waves

• Richard Weaver
• Physics, University of Illinois

Complex Systems May 2008
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Typical Experiment
Statistics?
Room Acoustics with ultrasound in solids
Object Size 10 cm Typical wavelength
6 mm
Signal Duration Mean free time between
scatterings 1/frequency 100 msec
30 msec
2 msec
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Conjecture Ultrasonics in High Q Reverberant
Bodies of generic shape (ray-chaotic?) is
described - for statistical purposes - by the
Gaussian Orthogonal Ensemble of Random Matrix
Theory In particular Correlations of their
eigenfrequencies Their sensitivity to
time-independent perturbations Their behavior
when dissipation is introduced
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Multiply scattering bodies
Ray-chaotic cavities
Systems described by Random Matrix Theory?
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Conventional Ultrasonics looks at direct signal
from one transducer to another at short times
Surface Rayleigh wave Surface skimming P wave
(P) Bottom reflected P wave (LL) P wave reflected
twice off bottom(LLLL)
Observe ray arrivals corresponding to
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Diffuse Field Ultrasonics looks at the coda
Looks like noise under a decaying envelope
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What do we do with the coda ?
The obvious Dissipation Diffusion / Transport
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Often, the behavior of Spectral Power Density
versus time is an
Exponential decay, at a rate related to internal
friction.
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Transport in media with random multiple scattering
fits well to a diffusion equation with dissipation
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Mesoscopic Phenomena that we study Level
Correlations Fidelity
decay Return probabilities / level width
statistics Coherent (enhanced)
Backscatter Anderson Localization
(2-d) (0-d) S-matrix
statistics Field-field correlations Many
phenomena whose study requires high Qs and
random scattering
todays talk
in that phases are maintained the response is
coherent wave interference must be
considered.
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Conjecture (Bohigas, Casati . .
1980) Systems whose classical mechanics is
chaotic have quantum mechanics in the ? ? 0
limit with statistics like those of RMT In
particular - its level correlations its
S-matrix statistics its distribution of decay
rates its fidelity under perturbations in the
Hamiltonian.
Supported by numerical experiments,
and microwave laboratory experiments,
mostly on 2-d "quantum billiards"
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I. Eigenfrequency Correlations
Conjecture Reverberant Elastic Wave Systems
- even if not obviously ray-chaotic will, for
practical purposes at finite wavelength, exhibit
the same accord with RMT
e.g.
or
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Spectrum of a rectangular block with planar
cut(s) pseudo integrable?
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pseudo integrable, and yet . . . .
Nearest neighbor level-spacing histogram is in
fine accord with RMT
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pseudo integrable, and yet . . . .
Number variance
Spectrum has the rigidity of RMT
This sort of behavior later seen in Quartz blocks
also
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Why? answers(?) Elastodynamics with
its diffractive scattering at corners and
edges spreads rapidly in phase
space Physical object is non-smooth ray
paths are chaotic
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Temperature Dependent Distortion (fidelity) of
Fields
II. Fidelity decay
Phys. Rev. Lett. 90, 254302 (2003)
See also Gorin, Prosen Seligman A RMT
formulation of fidelity decay, New J Phys 6,
(2004) Schafer, Gorin, Seligman and Stoeckman
Fidelity amplitude of scattering matrix in
microwave cavities Prosen, Seligman and
Znidaric, Theory of Quantum Loschmidt echoes
Prog Th Phys (2003) Gorin, Seligman and Weaver
Scattering fidelity in elastodynamics, Physical
Review E 73 015202 (2006)
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Close-up view of signal from two different
temperatures
Signal has a) shifted to the right (dilated
actually) and b) distorted slightly
because lower wave speeds at higher
temperatures because S and P waves dilate at
different rates
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Experiments
An initially heated sample is allowed to cool
slowly in a vacuum Temperature is
monitored. Waveforms are taken regularly.
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The signals S(t) from different Temperatures are
compared by forming a Normalized Cross
Correlation Function X(e) between two signals,
S1(t) and S2(t) taken at Temperatures T1 and
T2
X's maximum, at some mean dilation e is
analogous to 'fidelity'
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Distortion ( fidelity ) versus time for the
"medium block" Compared to one-parameter fit from
RMT
Phys. Rev. Lett. 90, 254302 (2003) Phys. Rev. E
73 015202 (2006)
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• Distortion ( fidelity ) versus time for the
  "rectangle"
• Compared to one-parameter fit from RMT
• Fidelity is weaker than in medium block,
• but fit to RMT is as good
• Even though object is
• regular
• symmetric (3 reflection symmetries)

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More recent measurements (under review) on medium
block One-parameter fits to RMT/GOE (?) and
Poisson(-----)
250 kHz
900 kHz
RMT again does a good job on this irregular
object,
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and measurements on rectangle . .
250 kHz
900 kHz
Again, fits to RMT (GOE) are superior.
Consistent with level correlation picture
elastic waves conform well to RMT, even in
systems that are not ray chaotic
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Non-exponential Dissipation in a Lossy
Elastodynamic Body,
III. Return probabilities / level width
statistics
Phys. Rev. Lett. 91 194101 (2003)
Thermodynamic argument At perfect coupling, each
channel carries power at a rate e Dw /2p
With M perfectly coupled channels, dissipation
rate from body will be M/THeisb
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• Two models for losses into a finite number of
  channels
• Breit-Wigner ( Porter-Thomas)
• Each mode overlaps each channel a random amount
  given by
• an assumed real Gaussian random process.
• E is then a product over channels c

2) RMT
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Comparison of RMT and Breit-Wigner model
For a case of four channels each coupled at 80
of maximum
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Measurements
At 250 kHz
Phys. Rev. Lett. 91 (2003)
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Fits to Theories
Below 580 kHz, there are 4 channels in the
wire two flexural (identically coupled) one
torsional one extensional Coupling strengths
are not known a priori
RMT 4 parameters Eo g1g2 g3, g4 gi 1
Porter-Thomas, 4 parameters Eo s1s2 s3, s4
1/THeisnb si 0
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Fits to Theories
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Non-exponential decays
Breit-Wigner fits adequately at almost all
frequencies But often has to claim all
channels are 100 coupled Breit-Wigner fits
inadequately on occasion, at strong coupling RMT
fits well at all frequencies, and gives
physically plausible coupling strengths Differenc
es are small, but statistically significant.
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In Summary
Diffuse Elastodynamics in reverberant bodies is
well described by RMT - even in apparently
regular systems! Level statistics
Fidelity decay Width statistics/decay
rates Other mesoscopic behaviors of such
waves are also seen Anderson localization Enha
nced backscatter Ericson fluctuations Field
-field correlations