Interstellar Levy Flights. Collaborators: Stas Boldyrev (U Chicago: Theory) Ramachandran (Berkeley: Pulsar Observing) Avinash Deshpande (Arecibo, Raman Inst: More Pulsars) Ben Stappers (Westerbork: Crucial pulsar person). Outline of Talk: Gambling with Pennies Statistics
Stas Boldyrev (U Chicago: Theory)
Ramachandran (Berkeley: Pulsar Observing)
Avinash Deshpande (Arecibo, Raman Inst: More Pulsars)
Ben Stappers (Westerbork: Crucial pulsar person)
Outline of Talk:
Gambling with Pennies
Application to California Lotto
Interstellar Games of Chance
You are given 1¢
Flip 1¢: win another 1¢ each time it comes up “heads”
Play 100 flips
You are given 2¢
Flip 1¢: double your winnings each time it comes up “heads”
You must walk away (keep winnings) when it comes up “tails” (100 flips max)Gamble and Win in Either Game!
Value = ∑ Probability($$)($$) = $0.50
… for both games
Note: for Cauchy, > $0.25 of “Value” is from payoffs larger than the US Debt.
The distributions of net outcomes will approach Levy stable distributions.
“Levy Stable” => when convolved with itself, produces a scaled copy.
In 1D, stable distributions take the form:
P b($)=∫ dk ei k ($) e-|k|
Gauss will approach a Gaussian distribution b=2
Cauchy will approach a Cauchy distribution b=1
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 ”Cauchy”, you must play enough times to win the top prize.Fine Print
…and win it many times
∑ Probability($$)($$)N≈(Top $)N
Are there observable effects?
Are there media where Cauchy is true?
Do observations tell us about the medium?
Isn’t this talk about insterstellar travel?When waves travel through random media,do they choose deflection angles via Gauss or Cauchy?
Simulated VLB Observation of Pulsar B1818-04
Excess flux at short baseline: big “halo”
Best-fit Gaussian model
*”Rotundate” baseline is scaled to account for elongation of the source (=anisotropic scattering).
It Has Already Been Done
Desai & Fey (2001) found that images of heavily-scattered sources in Cygnus did not resemble Gaussian distributions: they had a “cusp” and a “halo”.
Actually, radio interferometry measures the Fourier transform of the image -- usually confusing -- but convenient for Levy distributions!
Intrinsic structure of these complicated sources might create a “halo” – but probably not a “core”!
Mostly because paths with lots of bends take longer!
For Cauchy, many paths have only small deflections – and 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)
Observed pulses from pulsars tend to have sharper rise and more gradual fall than Gauss would predict.
Solid curve: Best-fit model b=1
Dotted curve: Best-fit model b=2
Horizontal lines show zero-level
(Because the scattering material is dispersive.)
From Westerbork: Ramachandran, Deshpande, Stappers, Gwinn
Pulse at 1230 MHz
(no one’s keeping track of time)
Pulse at 880 MHz
Fact: If the distribution for the steps has a power-law tail, then the result is not drawn from a Gaussian.It will approach a Levy distribution: P b(Dq)=∫ dk ei k Dq e-|k|P b(Dq) ®Gaussian for b®2Rare, large deflections dominate the path: a “Levy Flight”.
Klafter, Schlesinger, & Zumofen 1995, PhysToday
JP Nolan: http://academic2.american.edu/~jpnolan/