Lévy walk dynamics in an external harmonic potential

Abstract

Lévy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs, their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some long-standing puzzles around LWs.

13 pages, 5 figures, RevTeX

References in corpus (30)

• The scaling laws of human travel
• Anomalous transport in the crowded world of biological cells
• Lévy walks
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Langevin equations for continuous time Lévy flights
• Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins
• Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
• Lévy flights in random environments
• Relative dispersion in fully developed turbulence: The Richardson's Law and Intermittency Corrections
• From anomalous energy diffusion to Levy walks and heat conductivity in one-dimensional systems
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• Towards deterministic equations for Levy walks: the fractional material derivative
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics
• First-passage and first-hitting times of Levy flights and Levy walks
• Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk
• Levy flights in quenched random force fields
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Exact solution of a Levy walk model for anomalous heat transport
• Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
• Random time averaged diffusivities for Lévy walks
• Fractional Brownian motion with a reflecting wall
• Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices
• Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs and equipartition: Statistics of confined cold atoms
• Lévy walk revisited: Hermite polynomial expansion approach
• Levy walk with multiple internal states
• Universality in the Onset of Super-Diffusion in Lévy Walks
• L{é}vy walks on finite intervals: A step beyond asymptotics
• Deviations from generalized Equipartition in Confined, Laser Cooled Atoms

Cited by in corpus (6)

• Lévy-walk-like Langevin dynamics affected by a time-dependent force
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Lévy on-off intermittency
• Lévy walk dynamics in non-static media
• Underdamped, anomalous kinetics in double-well potentials
• Gaussian Process and Levy Walk under Stochastic Non-instantaneous Resetting and Stochastic Rest