Galaxies and Coffee Cups

One of the most striking features of the universe, as revealed by modern telescopes, is the clustering that's apparent in the sky. Made up of more than a billion stars, our galaxy, the Milky Way, is a member of a cluster of galaxies, and this local cluster, in turn, is a member of a larger aggregation known as a supercluster.

On even larger scales, clusters of clusters of galaxies appear to group into great walls, strings, sheets, and other structures, with unexpectedly large voids completely free of visible matter between these assemblages.

Interestingly, perfectly simple statistical rules can generate random collections of points that show the same self-similar clustering behavior and exhibit the same large voids evident in the distribution of galaxies in the universe. One such approach involves random walks ("Walking Wild I") in which the size of the steps are not fixed but vary in particular ways.

In the early part of the twentieth century, mathematician Paul Lévy (1886-1971) explored the possibilities and discovered a class of random walks in which steps vary in size, from infinitesimally small to infinitely large, so no average or characteristic length can be calculated.

The movements are different from Brownian motion ("Quivering Particles I") in that a Lévy walker takes steps of different lengths, with longer steps recurring proportionally less often than shorter steps. A jump ten times longer than another, for example, happens only one-tenth as often. It makes sense to call such excursions "flights."

In two dimensions, Lévy flights correspond roughly to a sequence of long jumps separated by what looks like periods of shorter ventures in different directions. Each "stopover," however, is itself made up of extended flights separated by clusters of short flights, and so on.

Magnifying any of the clusters or subclusters reveals a pattern that closely resembles the original, large-scale pattern, which means that Lévy flights have a fractal geometry—the parts on all scales closely resemble the whole.

A Lévy walker takes steps of different lengths, with longer steps occurring proportionally less often than shorter steps. The illustrated flight, in three dimensions, goes for 2,000 segments. The color of a step depends on its length. G.M. Viswanathan.

In two dimensions, the most striking visual difference between Brownian random walks and Lévy flights is the area they cover in a given time. Lévy flights sample a much larger territory than the corresponding Brownian random walks.

A comparison of a Lévy flight (left) with a Brownian random walk (right) reveals that, for the same number of steps, a Lévy flight covers a considerably larger area. The Brownian random walk lacks the long segments characteristic of a Lévy flight. G.M. Viswanathan.

A similar structure in games of chance governs the pattern of successive ruins, when the player loses everything yet is given the opportunity to continue (on credit) and lose again. The resulting Lévy distribution of the frequency of successive ruins is very different from the bell-shaped symmetry if a Gaussian distribution, which characterizes ordinary random walks.

In the Lévy case, the distribution is strongly skewed, with a very long tail that drops off to zero very slowly as the size of the events increases.

Mathematician Benoit B. Mandelbrot (1924-2010) originally learned about these different random walks from Lévy himself. Mandelbrot later extended and applied Lévy's ideas in his formulation of fractal geometry.

Mandelbrot found that he could use Lévy flights to create convincing portraits of the distribution of visible matter in the universe. He simply erased the long jumps and made each stopover represent a star, a galaxy, or some other blob of matter. The resulting pattern of clustered spots, each of which in turn is made up of subclusters, resembles the sheets, bubbles, and other aggregations of galaxies evident in astronomical observations.

Of course, Mandelbrot's model doesn't necessarily account for the way galaxies actually formed in the universe, but it does suggest the kind of structure that may be present.

Lévy flights and the statistics associated with them also provide useful models of turbulent diffusion. If you add a drop of cream to your coffee without unduly disturbing the liquid, the random motion of the molecules slowly spreads the cream into the coffee. Stirring, however, adds turbulence, and the liquids mix much more rapidly.

Mathematically, it's possible to think of turbulence as the combined effect of a large number of vortexes—whirlpools of all sizes and strengths. Any particles (or molecules of the constituents of cream) caught in such whirlpools would be rapidly separated from one another and dispersed. A plot of the changes in distance between two initially adjacent particles would look much more like a Lévy flight than a Brownian random walk.

Lévy flights can arise out of chaotic systems, in which a sensitive dependence on initial conditions plays a crucial role, and out of random systems, reflecting the same sort of haphazardness shown in Brownian motion. New statistics based on Lévy flights must be used to characterize these unpredictable phenomena. Such models may be useful for describing, for example, the transport of pollutants and the mixing of gases in Earth's atmosphere.

"In these complex systems, Lévy flights seem to be as prevalent as diffusion is in simpler systems," noted physicist Michael Shlesinger, who pioneered the applications of Lévy statistics to turbulent diffusion and other physical phenomena ("Time to Relax II"). (See An Unbounded Experience in Random Walks with Applications by Michael F. Shlesinger.)

In both the atmosphere and the ocean, fractal patterns associated with turbulence may have a strong influence on ecosystems, affecting, for example, the foraging behavior of certain birds. Weather systems and the distribution of plankton, krill, and other organisms in the ocean may guide the flight patterns of the wandering albatross.

Next: Flights of the Albatross