Levy Flights in Inhomogeneous Media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy 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.Comment: 4 pages, 4 figure

Anomalous diffusion as a signature of collapsing phase in two dimensional self-gravitating systems

A two dimensional self-gravitating Hamiltonian model made by $N$ fully-coupled classical particles exhibits a transition from a collapsing phase (CP) at low energy to a homogeneous phase (HP) at high energy. From a dynamical point of view, the two phases are characterized by two distinct single-particle motions : namely, superdiffusive in the CP and ballistic in the HP. Anomalous diffusion is observed up to a time $\tau$ that increases linearly with $N$. Therefore, the finite particle number acts like a white noise source for the system, inhibiting anomalous transport at longer times.Comment: 10 pages, Revtex - 3 Figs - Submitted to Physical Review

Particle Dispersion on Rapidly Folding Random Hetero-Polymers

We investigate the dynamics of a particle moving randomly along a disordered hetero-polymer subjected to rapid conformational changes which induce superdiffusive motion in chemical coordinates. We study the antagonistic interplay between the enhanced diffusion and the quenched disorder. The dispersion speed exhibits universal behavior independent of the folding statistics. On the other hand it is strongly affected by the structure of the disordered potential. The results may serve as a reference point for a number of translocation phenomena observed in biological cells, such as protein dynamics on DNA strands.Comment: 4 pages, 4 figure

Statistics of resonances and of delay times in quasiperiodic Schr"odinger equations

We study the statistical distributions of the resonance widths ${\cal P} (\Gamma)$, and of delay times ${\cal P} (\tau)$ in one dimensional quasi-periodic tight-binding systems with one open channel. Both quantities are found to decay algebraically as $\Gamma^{-\alpha}$, and $\tau^{-\gamma}$ on small and large scales respectively. The exponents $\alpha$, and $\gamma$ are related to the fractal dimension $D_0^E$ of the spectrum of the closed system as $\alpha=1+D_0^E$ and $\gamma=2-D_0^E$. Our results are verified for the Harper model at the metal-insulator transition and for Fibonacci lattices.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let

What determines the spreading of a wave packet?

The multifractal dimensions D2^mu and D2^psi of the energy spectrum and eigenfunctions, resp., are shown to determine the asymptotic scaling of the width of a spreading wave packet. For systems where the shape of the wave packet is preserved the k-th moment increases as t^(k*beta) with beta=D2^mu/D2^psi, while in general t^(k*beta) is an optimal lower bound. Furthermore, we show that in d dimensions asymptotically in time the center of any wave packet decreases spatially as a power law with exponent D_2^psi - d and present numerical support for these results.Comment: Physical Review Letters to appear, 4 pages postscript with figure

A mechanical model of normal and anomalous diffusion

The overdamped dynamics of a charged particle driven by an uniform electric field through a random sequence of scatterers in one dimension is investigated. Analytic expressions of the mean velocity and of the velocity power spectrum are presented. These show that above a threshold value of the field normal diffusion is superimposed to ballistic motion. The diffusion constant can be given explicitly. At the threshold field the transition between conduction and localization is accompanied by an anomalous diffusion. Our results exemplify that, even in the absence of time-dependent stochastic forces, a purely mechanical model equipped with a quenched disorder can exhibit normal as well as anomalous diffusion, the latter emerging as a critical property.Comment: 16 pages, no figure

Quenched and Negative Hall Effect in Periodic Media: Application to Antidot Superlattices

We find the counterintuitive result that electrons move in OPPOSITE direction to the free electron E x B - drift when subject to a two-dimensional periodic potential. We show that this phenomenon arises from chaotic channeling trajectories and by a subtle mechanism leads to a NEGATIVE value of the Hall resistivity for small magnetic fields. The effect is present also in experimentally recorded Hall curves in antidot arrays on semiconductor heterojunctions but so far has remained unexplained.Comment: 10 pages, 4 figs on request, RevTeX3.0, Europhysics Letters, in pres

A repulsive trap for two electrons in a magnetic field

We study numerically and analytically the dynamics of two classical electrons with Coulomb interaction in a two dimensional antidot superlattice potential in the presence of crossed electric and magnetic fields. It is found that near one antidot the electron pair can be trapped for a long time and the escape rate from such a trap is proportional to the square of a weak electric field. This is qualitatively different from the case of noninteracting electrons which are trapped forever by the antidot. For the pair propagation in the antidot superlattice we found a broad parameter regime for which the pair is stable and where two repulsive electrons propagate together on an enormously large distance.Comment: revtex, 5 pages, 6 figure

Experimental evidence for the role of cantori as barriers in a quantum system

We investigate the effect of cantori on momentum diffusion in a quantum system. Ultracold caesium atoms are subjected to a specifically designed periodically pulsed standing wave. A cantorus separates two chaotic regions of the classical phase space. Diffusion through the cantorus is classically predicted. Quantum diffusion is only significant when the classical phase-space area escaping through the cantorus per period greatly exceeds Planck's constant. Experimental data and a quantum analysis confirm that the cantori act as barriers.Comment: 19 pages including 9 figures, Accepted for publication in Physical Review E in March 199

Equilibrium and dynamical properties of two dimensional self-gravitating systems

A system of N classical particles in a 2D periodic cell interacting via long-range attractive potential is studied. For low energy density $U$ a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy U_c. A theoretical analysis within the canonical ensemble identifies such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near $U_c$. The dynamical behaviour of the system is affected by this transition : below U_c anomalous diffusion is observed, while for U > U_c the motion of the particles is almost ballistic. In the collapsed phase, finite $N$-effects act like a noise source of variance O(1/N), that restores normal diffusion on a time scale diverging with N. As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N \to \infty. A Lyapunov analysis reveals that for U > U_c the maximal exponent \lambda decreases proportionally to N^{-1/3} and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear non ergodicity of the system, a common scaling law \lambda \propto U^{1/2} is observed for any initial conditions.Comment: 17 pages, Revtex - 15 PS Figs - Subimitted to Physical Review E - Two column version with included figures : less paper waste