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)

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

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

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!

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

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

•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

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?

• 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.

• 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?

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 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”)

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

“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.

“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.

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

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

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

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).

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

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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and visitors to Inst Theor Phys.• Includes: competitive salary & benefits, plus substantial budget for

research expenses.

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