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Interstellar Levy Flights

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:

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Interstellar Levy Flights

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  1. 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. “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. 2 Games of Chance Value = ∑ Probability($$)($$) = $0.50 … for both games Note: for Levy, > $0.25 of “Value” is from payoffs larger than the total US Debt.

  3. 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 Moments of the Games

  4. 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. But Everything Becomes Gaussian! …and win it many times (>>1025) plays.

  5. 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 b=2 Levy will approach a Cauchy or Lorentzianb=1 b

  6. Mantegna & Stanley, Nature 1995 Probability 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

  7. 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.

  8. 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?

  9. Doesn’t 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.

  10. Kolmogorov orLevy – 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.

  11. 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)

  12. 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”)

  13. 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.

  14. “Traditional” Kolmogorov: • Pulse Broadening: tµl2a/(a-2)d1+4/2(a-2) tµl4.4d2.2, for a=11/3 Levy Flight (Kolmogorov): • Pulse Broadening: tµl2a/(a-2)d1+4/ b(a-2) tµl4.4d4, 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 statistics.

  15. “Traditional” Kolmogorov: • Pulse Broadening: tµl2a/(a-2)d1+4/2(a-2) tµl4.4d2.2, for a=11/3 Levy Flight (Kolmogorov): • Pulse Broadening: tµl2a/(a-2)d1+4/ b(a-2) tµl4.4d4, 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.

  16. 2. Williamson 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: b=2 Solid line: b =1 Dashed line: b =2/3 (Scaled to the same maximum and width at half-max)

  17. PSR 1818-1422 • Williamson (1975) found thin screens reproduced pulse shapes better than an extended medium (b=2). • Levy works about as well as a thin-screen model--work continues. Solid curve: Best-fit model b=1 Dotted curve: Best-fit model b=2 Both: Extended, homogeneous medium • Fits to data must include offsets & scales in amplitude and time, as well as effects of quantization.

  18. 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 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.

  19. 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). 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”!

  20. 31/2. Fiedler Extreme Scattering Events, Parabolic Arcs in Secondary Spectra, Intra-Day Variability, and similar phenomena suggest occasional scattering to very large angles. Random Deterministic • 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?

  21. 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

  22. – Advertisement – 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. TABASGO Prize Graduate Fellowships in Astrophysics at UC Santa Barbara • 2 years fellowship support

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