Interstellar Levy Flights

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Interstellar Levy Flights
C.R. Gwinn (UC Santa Barbara)
Collaborators
Levy flights and Turbulence Theory Stas Boldyrev
(U Chicago ? Univ Wisconsin )
Papers ApJ, Phys Rev Lett 2003, 2004, also
astro-ph Or Search Levy Flights on Google for
our page (2nd from top)
Pulsars Ben Stappers (Westerbork Crucial
pulsar person) Avinash Deshpande (Raman Inst
More Pulsars)
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2 Games of Chance

• Gauss
• You are given 0.01
• Flip coin win another 0.01 each time it lands
  heads up
• Play 100 flips

• Levy
• You are given 0.02
• Guess 25 digits 0-9.
• Multiply your winnings 11? for each successive
  correct digit.

Value ? Probability()?() 0.50
for both games
Note for Levy, gt 0.25 of Value is from
payoffs larger than the total US Debt.
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Moments of the Games

• For Gauss
• M1, M2 completely characterize the game.

• For Levy
• Higher moments Ngt1 are (almost) completely
  determined by the top prize
• MN10-25?(1026)N

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But Everything Becomes Gaussian!
and win it many times (gtgt1025) plays.
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Attractors
After many plays the distribution of outcomes
will (usually) approach a Levy-stable
distribution.
In one dimension, symmetric Levy-stable
distributions take the form Pb()? dk eik?
e-k If games are made zero-mean Gauss will
approach a Gaussian distribution b2 Levy will
approach a Cauchy or Lorentzian b1
b
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Example Stock markets follow (nearly) Levy
statistics rather than Gaussian statistics. This
is critical to pricing of financial
derivatives. See J. Voit Statistical Mechanics
of Finance
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Scattering is 2D

• In 2 or higher dimensions, Levy-stable
  distributions can have many forms.
• They are not always easy to visualize or
  classify.
• Results here are for 2D analogs of the 1D
  symmetric Levy-stable distributions.

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Are deflection angles for interstellar wave
propagation chosen by Gauss or Levy?

• Theory usually assumes Gauss.

Are there observable differences? Are there
media where Levy is true? Can statistics depend
on physics of turbulence?
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?Doesnt the Kolmogorov Theory fully describe
turbulence?

• Kolmogorov predicts scaling for velocity
  difference with separation
• ?v? ?x1/3 (with corrections for higher
  moments)
• Density differences ?n can follow related
  scaling.
• The distribution of density differences P(Dn) may
  be either Gaussian or Levy.
• A Levy pdf for P(Dn) leads to a Levy flight.

?Kolmogorov Levy may coexist.
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Kolmogorov or Levy or Both?

• Intermittency in turbulence involves important,
  rare events (as in Kolmogorovs later work and
  She-Levesque scaling law).
• Although large but rare events also dominate
  averages in Levy flights, the resulting
  distributions are not described by moments, as in
  these theories.
• Many scenarios can give rise to Levy flights
• For example, deflection by a series of randomly
  oriented interfaces (via Snells Law) yields b1
• Interestingly, Kolmogorov co-authored a book on
  Levy-stable distributions, with theorems on
  basins of attraction.

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Parabolic Wave Equation
Parabolic wave equation takes the usual form,
with Levy distribution for the random
term. Approaches to solution Ray-tracing via
Pseudo-Hamiltonian formalism (Boldyrev CG ApJ
2003) Find 2-point coherence function via
transform of superposed screens (Boldyrev CG
PRL 2003, ApJ 2004)
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31/2 Observable Consequencesfor Gauss vs Levy

• Scaling of pulse broadening with distance
  (Sutton Paradox)
• Shape of a scattered pulse (Williamson Paradox)
• Shape of a scattered image (Desai Paradox)
• Extreme scattering events (Fiedler Events)

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1. Sutton
Pulses must broaden like (distance)2      lt q2gt
? d      t ? lt q2gt d But measurements show
t ? (distance)4
To resolve the paradox, Sutton (1974) invoked
rare, large events the probability of
encountering much stronger scattering material
increases dramatically with distance.
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Levy Flights can rephrase the nonstationary
statistics invoked by Sutton, as stationary,
non-Gaussian statistics. Suitable choice for b
yields the observed scaling with distance and
wavelength, with Kolmogorov statistics.
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Levy Flights can rephrase the nonstationary
statistics invoked by Sutton, as stationary,
non-Gaussian statistics. Suitable choice for b
yields the observed scaling with distance and
wavelength, with Kolmogorov scaling.
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Gauss and Levy predict different impulse-response
functions for extended media
2. Williamson
For Levy, most paths have only small delays but
some have very large ones relative to Gauss
Dotted line b2 Solid line b 1 Dashed line b
2/3
(Scaled to the same maximum and width at half-max)
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PSR 1818-1422

• Williamson (1975) found thin screens reproduced
  pulse shapes better than an extended medium
  (b2).
• Levy works about as well as a thin-screen
  model--work continues.

Solid curve Best-fit model b1 Dotted curve
Best-fit model b2 Both Extended, homogeneous
medium

• Fits to data must include offsets scales in
  amplitude and time, as well as effects of
  quantization.

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3. Desai
Lets Measure the Deflection by Imaging!

• At each point along the line of sight, the wave
  is deflected by a random angle.
• Repeated deflections should converge to a
  Levy-stable distribution of scattering angles.

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It Has Already Been Done
Desai Fey (2001) found that images of some
heavily-scattered sources in Cygnus did not
resemble Gaussian distributions they had a
cusp and a halo.
Intrinsic structure of these sources might
contribute a halo around a scattered image
but probably could not create a sharp cusp!
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• Can these events be described statistically?
• Are Levy statistics appropriate?
• Could these join typical scattering in a single
  distribution?
• Might they be localized in a particular phase of
  the ISM?

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Summary

• Sums of random deflections can converge to
  Levy-stable distributions.
• b parametrizes some of these, including Gaussian.
• Propagation through random media with
  non-Gaussian statistics can result in Levy
  flights.
• Observations can discriminate among various Levy
  models for scattering
• DM-vs-t
• Pulse Shape
• Scattering disk structure
• Rare scattering to large angles (?)
• Extreme scattering events
• Parabolic Arcs in Secondary Scintillation Spectra
• Intra-Day Variability

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TABASGO Prize Postdoctoral Fellowship in
Astrophysics at UC Santa Barbara
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• Primary qualification Promise of independent
  research excellence.
• May work independently or with UCSB faculty,
  postdocs, students and visitors to Inst Theor
  Phys.
• Includes competitive salary benefits, plus
  substantial budget for research expenses.

TABASGO Prize Graduate Fellowships in
Astrophysics at UC Santa Barbara

• 2 years fellowship support

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