EXPLORABLES

by Dirk Brockmann

This explorable illustrates the properties of a class of random walks known as Lévy flights. To get the most out of this explorable, you may want to check out the explorable Albert & Carl Friedrich on ordinary random walks and diffusion first.

Lévy flights, unlike ordinary random walks, are superdiffusive and scale-free, their paths have a self-similar structure. They play a role in numerous systems in physics but also in ecology and other fields.

Press play and keep on reading.

This is how it works

The two-dimensional Lévy flights here all start at the origin and perform a sequence of random steps. At every iteration \(n=1,2,3,...\) their position \(\mathbf{R}_{n}=(X_n,Y_n)\) is updated by a displacement \(\mathbf{r}_n=(x_n,y_n)\) so that

\[ \mathbf{R}_{n+1}=\mathbf{R}_{n}+\mathbf{r}_n. \]

At every step a random direction is chosen (uniformly) and a step-length \(r\) is drawn from a probability density that follows an inverse power-law

\[ p(r)\propto\frac{1}{r^{1+\mu}} \]

for \(r>r_0\) (\(r_0\) is some minimal step length). So short jumps are more frequent than long ones. The exponent \(0<\mu\) is the Lévy exponent and a defining feature of the geometry of the trajectories. You can pick different values for \(\mu\) in the control panel. You can also choose the number of walks to be generated during a run (A run automatically stops after 1000 steps).

As the walks proceed, the path, the visited locations and the current position are displayed for each walk. You can use the toggles to display or hide locations, current position, and paths.

Observe this

When you choose a Lévy flight, say for \(\mu=1\), you will see that a walker remains in some location for some time making many short jumps. But every now and then a long jump occurs, taking the walker to a distance place. As we zoom out, longer and longer jumps occur, turning the path of the walker into a self-similar structure. When you hide the path and just look at the visited locations, this feature is best visible. For smaller exponents the "dust" of visited locations becomes more and more dilute. It turns out that the visited locations form a random fractal of dimension \(\mu\) if \(\mu<2\).

What's up with the central limit theorem?

You may rightfully ask why Lévy flights appear to have such different geometry than paths of ordinary random walks / diffusion processes? After all, does not the central limit theorem dictate that asymptotically these walks should resemble trajectories of ordinary diffusion as illustrated in the explorable Albert & Carl Friedrich (check it out, if you haven't done so already)? Well, for Lévy flights, one of the conditions required for the central limit theorem is violated for \(\mu\leq 2\), namely that the variance of the step length \(r\) is finite. In fact, for the power-law \(p(r)\) above the variance is infinite when \(\mu\leq 2\).

Superdiffusion

As a consequence of this, the position after \(n\) steps no longer exhibits the typical scaling \(R_n\sim \sqrt{n}\). Instead, Lévy flights are superdiffusive and their typical distance from the origin scales as

\[ R_n \sim n^{\frac{1}{\mu}} \]

Lévy flights, just like ordinary random walks have a limiting distribution, only that it is no longer a Gaussian function. One can show that for \(\mu < 2 \) the scaled variable \(\mathbf{z}_{n} = \mathbf{R}_n /n^{\frac{1}{\mu}}\) has a probability density that approaches a function \(L_\mu(z)\) given by

\[ L_\mu(z)=\int d\mathbf{k}e^{-|\mathbf{k}|^\mu-i\mathbf{k}\mathbf{x}}. \]

When \(\mu>2\) the variance of the single steps is finite and everything is back to normal.


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