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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)
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, > $0.25 of “Value” is from payoffs larger than
the total US Debt.
Moments of the Games
MN=∑ Probability($$)($$)N
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
But Everything Becomes Gaussian!
The Central Limit Theorem says: the outcome
will be drawn from a Gaussian distribution,
centered at N$0.50, with variance given by….
To reach that limit with ”Levy”,
you must play enough times to
win the top prize.
…and win it many times
(>>1025) plays.
Attractors
After many plays: the distribution of outcomes will
(usually) approach a Levy-stable distribution.
In one dimension, symmetric Levy-stable distributions
take the form:
Pb($)=∫ dk
eik$
b
-|k|
e
If games are made zero-mean:
Gauss will approach a Gaussian distribution b=2
Levy will approach a Cauchy or Lorentzian b=1
Probability
Mantegna & Stanley, Nature 1995
Change in Standard &Poors 500 Index, Dt=1 min
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
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.
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?
Doesn’t the Kolmogorov Theory
fully describe turbulence?
• Kolmogorov predicts scaling for velocity
difference with separation:
Dv Dx1/3 (with corrections for higher moments)
Density differences Dn 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.
Kolmogorov or Levy – or Both?
• 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 b=1
Interestingly, Kolmogorov co-authored a book on Levy-stable
distributions, with theorems on basins of attraction.
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)
31/2 Observable Consequences
for 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”)
1. Sutton
Pulses must broaden like
(distance)2:
< q2>  d
t  < q2> 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.
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.
“Traditional” Kolmogorov:
Levy Flight (Kolmogorov):
• Pulse Broadening:
t2a /(a-2) d1+4/2(a-2)
t4.4 d2.2, for a=11/3
• Pulse Broadening:
t2a /(a-2) d1+4/ b(a-2)
t4.4 d4, for a=11/3, b=4/5
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.
“Traditional” Kolmogorov:
Levy Flight (Kolmogorov):
• Pulse Broadening:
t2a /(a-2) d1+4/2(a-2)
t4.4 d2.2, for a=11/3
• Pulse Broadening:
t2a /(a-2) d1+4/ b(a-2)
t4.4 d4, for a=11/3, b=4/5
2. Williamson
Gauss and Levy predict different impulseresponse functions for extended media
Dotted line: b=2
Solid line: b =1
Dashed line: b =2/3
(Scaled to the same maximum and width at half-max)
For Levy,
most paths
have only
small delays –
but some have
very large
ones – relative
to Gauss
PSR 1818-1422
Williamson (1975)
found thin screens
reproduced pulse shapes
better than an extended
medium (b=2).
 Levy works about as
Solid curve: Best-fit model b=1
well as a thin-screen
Dotted curve: Best-fit model b=2
model--work continues.
Both: Extended, homogeneous medium
Fits to data must include offsets
& scales in amplitude and time, as
well as effects of quantization.
3. Desai
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.
b=1
b=2
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
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”.
Excess flux at long
baseline: sharp “cusp”
Excess flux at short
baseline: big “halo”
Best-fit Gaussian model
*”Rotundate” baseline is scaled to account for anisotropic scattering (see Spangler 1984).
Intrinsic structure of these sources might contribute a “halo” around a
scattered image – but probably could not create a sharp “cusp”!
31/2. Fiedler
Extreme 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
Random
Deterministic
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?
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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