Levy Flights and Superdiffusion

Now consider a particle walking randomly in a hemocoel being jiggled by the convective circulation of hemolymph as the hemocoel itself jiggles during flight. Our particle starts at one position, and takes steps in random directions. As we know, random walks may occur in all three dimensions depending on local conditions.

In some cases the jiggling may make the steps randomly long as well. Now suppose a 'walker' particle diffusing in the hemocoel jiggles as our bee flies. Now perhaps motion from the jigging combines with convection from normal circulation to add random velocity component vectors that force the random walker to pause for random amounts of time in between steps.

Normal Random Walk

In the hemocoel perhaps 'most' random walks spread out diffusing normally, the variance of a group of particles growing linearly over time. This variance describes the size of a typical group of diffusing particles. (Remember: average of the squares of the distance a random walker moves minus the square of the average of the distance the random walker moves.) Our diffusion constant is the rate at which the variance grows. Remember D, the diffusion constant, is large when particles move faster in water and smaller when they move slower in syrup.

For cases where a particle's step length is random, D depends on the average squared step length rather than the length of an average step. Also, if the random walker takes one step every two seconds, it is reasonable to guess that D might be smaller.

Levy flight

Now what happens when the average squared step is very large compared with the usual step length or, we might say, infinite? Such a walk with very large segments are Levy flights, and D becomes infinite.

The two pictures compare a normal random walk and a Levy flight. The thousand steps adding up to each particle's trajectory are of random lengths. In the normal walk the probability of a long step is proportional to L(-3 8). In the Levy flight, the probability of a long step is proportional to L(-2 2), which is more probable than the normal case.

Follow a normal random walk for a very long time to begin seeing 'normal' behavior, that is we can no longer make out the small steps. However, the average lengths of all the steps taken together determine the trajectory's pattern. In a Levy flight, convection and jiggling create the long, infrequent steps, the flights. These long straight segments determine the trajectory's pattern. In a Levy flight, the few, long rare steps, the flights, mostly determine a walker's position. Thus, in a Levy flight, the individual steps do not average out.

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