Lévy Flights in Inhomogeneous Media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Lévy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Lévy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Lévy index $\mu$. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks.

Figure 1

A particle jumps randomly along a folded copolymer which consists of two types of periodically arranged monomers. Each monomer type has a different potential sketched qualitatively in the top left. Because of conformational changes of the chain, the particle can jump to a site close in Euclidean space yet distant in chemical coordinate $x$.

Figure 2

Band structure ${\kappa }_{n,q}$ as a function of $\beta$ for four different potentials. (A) A simple cosine potential $V\propto \cos (x/\lambda )$, (B) the square wave potential, and potentials given by $V\propto \pm [1+\cos (x/\lambda ){]}^{\gamma }$. When $\gamma \gg 1$, the latter possess localized high potential barriers (C) or localized potential wells (D). A Lévy flight ($\mu =1/2$) is compared to ordinary diffusion ($\mu =2$).

Figure 3

The quantity $g_{\mu }(\lambda /L)$ in the high temperature regime as a function of Lévy index $\mu$ and fixed system size (left). The solid lines (symbols) depict the results obtained from perturbation theory (numerics). Thick gray lines indicate the asymptotic limit of $1/4$ (1) if $\mu <2$ ($\mu =2$). Viewed as a function of inverse relative system size $\lambda /L$ for a set of values of $\mu$ (right) indicates that convergence to the limiting values is slowest when $\mu$ is small or slightly less than 2.

Figure 4

The generalized diffusion coefficient $D(\beta )$ as a function of inverse temperature for a chosen set of Lévy indices (indicated by the symbols in the lower left) and the set of potentials depicted in the upper right. The data were obtained by numerical diagonalization of (10). The dotted ($\mu <2$) and dashed ($\mu =2$) curves indicate the result obtained by the perturbation expansion for small $\beta$.