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-ph Or: 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”: M 1, M 2 completely characterize the game. For “Levy”: Higher moments N>1 are (almost) completely determined by the top prize: M N ≈  $10 26 ) N M N =∑ Probability($$)  $$) N

To reach that limit with ”Levy”, you must play enough times to win the top prize. …and win it many times (>>10 25 ) 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 e ik  $ 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 Probability 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   x 1/3 (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)

3 1/2 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”)

Pulses must broaden like (distance) 2 :  d   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)  d 1+4  2)  4.4  d 2.2, for  =11/3 Levy Flight (Kolmogorov): Pulse Broadening:    2)  d 1+4  2) d 4  4.4  d 4, 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)  d 1+4  2)  4.4  d 2.2, for  =11/3 Levy Flight (Kolmogorov): Pulse Broadening:    2)  d 1+4  2) d 4  4.4  d 4, 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:  =2 Solid line:  =1 Dashed 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  =1 Dotted curve: Best-fit model  =2 Both: Extended, homogeneous medium  Fits to data must include offsets & scales in amplitude and time, as well as effects of quantization. PSR

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 B  =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).

3 1/2. 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 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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