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Interstellar Levy Flights
Levy flights and Turbulence Theory:Stas Boldyrev (U Chicago Univ Wisconsin )
Collaborators:
Pulsars:Ben Stappers (Westerbork: Crucial pulsar person)Avinash Deshpande (Raman Inst: More Pulsars)
Papers: ApJ, Phys Rev Lett 2003, 2004, also astro-phOr: Search “Levy Flights” on Google for our page
(≈2nd from top)
C.R. Gwinn (UC Santa Barbara)
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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
Note: for Levy, > $0.25 of “Value” is from payoffs larger than the total US Debt.
… for both games
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Moments of the Games
For “Gauss”:
M1, M2 completely characterize the game.
For “Levy”:
Higher moments N>1 are (almost) completely determined by the top prize:
MN≈10-25$1026)N
MN=∑ Probability($$)$$)N
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To reach that limit with ”Levy”, you must play enough times to win the top prize.
…and win it many times (>>1025) plays.
The Central Limit Theorem says: the outcome will be drawn from a Gaussian distribution, centered at N$0.50, with variance given by….
But Everything Becomes Gaussian!
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After many plays: the distribution of outcomes will (usually) approach a Levy-stable distribution.
Attractors
In one dimension, symmetric Levy-stable distributions
take the form:
P($)=∫ dk eik$ e-|k|
If games are made zero-mean:
Gauss will approach a Gaussian distribution =2
Levy will approach a Cauchy or Lorentzian =1
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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
Change in Standard &Poors 500 Index, t=1 min
Pro
babi
lity
Mantegna & Stanley, Nature 1995
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•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.
Scattering is 2D
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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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• 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(n) may be either Gaussian or Levy.
• A Levy pdf for P(n) leads to a Levy “flight.”
Doesn’t the Kolmogorov Theory fully describe turbulence?
Kolmogorov & Levy may coexist.
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• Intermittency in turbulence involves important, rare events (as in Kolmogorov’s 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
Snell’s Law) yields =1
Interestingly, Kolmogorov co-authored a book on Levy-stable distributions, with theorems on basins of attraction.
Kolmogorov or Levy – or Both?
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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
1. Scaling of pulse broadening with distance (“Sutton Paradox”)
2. Shape of a scattered pulse (“Williamson Paradox”)
3. Shape of a scattered image (“Desai Paradox”)? Extreme scattering events (“Fiedler Events”)
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Pulses must broaden like (distance)2:
< 2> d < 2> d
But measurements show (distance)4
To resolve the paradox, Sutton (1974) invoked rare, large events: the probability of encountering much stronger scattering material increases dramatically with distance.
1. Sutton
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“Traditional” Kolmogorov:
• Pulse Broadening: 2)d1+42)
4.4d2.2, for =11/3
Levy Flight (Kolmogorov):
• Pulse Broadening: 2)d1+42)
4.4dd44, for =11/3, =4/5
Levy Flights can rephrase the nonstationary statistics invoked by Sutton, as stationary, non-Gaussian statistics.
Suitable choice for yields the observed scaling with distance and wavelength, with Kolmogorov statistics.
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“Traditional” Kolmogorov:
• Pulse Broadening: 2)d1+42)
4.4d2.2, for =11/3
Levy Flight (Kolmogorov):
• Pulse Broadening: 2)d1+42)
4.4dd44, for =11/3, =4/5
Levy Flights can rephrase the nonstationary statistics invoked by Sutton, as stationary, non-Gaussian statistics.
Suitable choice for 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
For Levy, most paths have only small delays – but some have very large ones – relative to Gauss
Dotted line: =2Solid line: =1Dashed line: =2/3
(Scaled to the same maximum and width at half-max)
2. Williamson
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Williamson (1975) found thin screens reproduced pulse shapes better than an extended medium (=2).
Levy works about as well as a thin-screen model--work continues.
Solid curve: Best-fit model =1Dotted curve: Best-fit model =2Both: Extended, homogeneous medium
Fits to data must include offsets & scales in amplitude and time, as well as effects of quantization.
PSR 1818-1422
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Let’s 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.
Probability(of deflection angle) –is– the observed image*.
* for a scattered point source.
Observations of a scattered point source should tell the distribution. Simulated VLB Observation of
Pulsar B1818-04
=1
=2
3. Desai
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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”.
It Has Already Been Done
Intrinsic structure of these sources might contribute a “halo” around a scattered image – but probably could not create a sharp “cusp”!
Best-fit Gaussian model
Excess flux at long baseline: sharp “cusp”
Excess flux at short baseline: big “halo”
*”Rotundate” baseline is scaled to account for anisotropic scattering (see Spangler 1984).
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31/2. FiedlerExtreme Scattering Events, Parabolic Arcs in Secondary Spectra, Intra-Day Variability, and similar phenomena suggest occasional scattering to very large angles.
•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?
DeterministicRandom
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Summary$ Sums of random deflections can converge: to Levy-stable
distributions. 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- 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• 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.
– Advertisement –
TABASGO Prize Graduate Fellowships in Astrophysics at
UC Santa Barbara• 2 years fellowship support
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