Interstellar Levy Flights

Levy Flights and Random Walks: A random walk is a path composed of many independent random steps. The path of one atom of argon in the Earth's atmosphere is a random walk; so is the price of a stock (at least in part, according to Burton Malkiel).

Typically, one thinks of a random walk as the result of many small, random steps. If a number of random walkers start from the same point, their final locations will typically follow a Gaussian distribution: the famous bell-shaped curve. In fact, the celebrated Central Limit Theorem says that the outcome will be Gaussian even if the steps aren't Gaussian! Right?? Not quite!

The result isn't always a Gaussian distribution. Some types of random series of events are dominated by rare, large events; these lead to non-Gaussian distributions of the results. An example is the lottery, where I expect that the one or two largest prizes will dominate my winnings, whether I buy 10 tickets or 10000. Another example is the stock market, where most of a stock's movement, over many years, often takes place on the best or worst few days. Mathematically speaking, the Central Limit Theorem holds only if the second moment of the probability distribution function for the steps is finite.

Levy distributions result when the second moment isn't finite. If the random walkers choose steps from a Levy distribution, their final locations will trace out the same Levy distribution. Such random walks are called "Levy flights" because of the occasional very large steps.

Interstellar Scattering: Radio waves traveling through interstellar space take paths that are random to some degree. Tenuous ionized gas in interstellar space deflects radio waves from direct paths. The effect is closely analogous to the deflection of light in convecting air, which causes the shimmering effect over a heater, and causes the stars to twinkle at night. It is also analogous to the deflection of water waves by shallow reefs in deep water, an effect well known to surfers.

The simplest picture for radio-wave scattering in the interstellar medium is a random walk: at each step, a ray is deflected by a random angle. One can view that angle as the result of deflection by a prism, with the prism angle and orientation random. If enough prisms are superposed, then the path is a random walk.

Interstellar Levy Flights: Stas Boldyrev, Carl Gwinn, Ramesh Bhat, and Conrad Hirano have been investigating the possibility that radio-wave propagation in the interstellar medium has statistics of a Levy Flight. In other words, rare, large deflections dominate the averages. More precisely, Levy flights can result if the differences in density between two points in interstellar space are drawn from a Levy distribution. (Of course, that distribution can still scale with the separation of the points: for example, according to a power-law with the Kolmogorov index. So Kolmogorov turbulence can create Levy flights.)

We are working to understand the predictions of the Levy picture for observations of scintillation and angular scattering. Some theories of super-sonic or super-alfvenic turbulence predict Levy-like distributions. For example, a collection of randomly oriented density steps (Burger's turbulence) will produce a Levy distribution with beta=1 (also known as a Cauchy or Lorentzian distribution). For pulsars, Levy statistics can explain the multipath broadening of pulses, as a function of the integrated electron density along the line of sight (the "tau-DM plot" for pulsars, sometimes known as the "elbow diagram"). This provides a statistically stationary explanation, in contrast to the non-stationary explanation of Sutton.

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Carl Gwinn
Last Modified: 24 Sept 2003