Random time averaged diffusivities for Lévy walks

Abstract

We investigate a Lévy-Walk alternating between velocities $\pm v_0$ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is $<x^2>\propto t^2$, the latter to enhanced diffusion with $<x^2>\propto t^{\nu }$, $1<\nu <2$. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.

References in corpus (4)

• The scaling laws of human travel
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

Cited by in corpus (31)

• Lévy walks
• Search with home returns provides advantage under high uncertainty
• First-passage and first-hitting times of Levy flights and Levy walks
• Quantifying the non-ergodicity of scaled Brownian motion
• Invariants of motion with stochastic resetting and space-time coupled returns
• Superdiffusive motion of membrane-targeting C2 domains
• Asymptotic densities of ballistic Lévy walks
• Scaling Green-Kubo relation and application to three aging systems
• Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes
• Anomalous diffusion in a quenched-trap model on fractal lattices
• Strange kinetics of bulk-mediated diffusion on lipid bilayers
• Optimization and universality of Brownian search in quenched heterogeneous media
• Phase Diagram in Stored-Energy-Driven Lévy Flight
• Scale invariant Green-Kubo relation for time averaged diffusivity
• A no-go theorem for ergodicity and an Einstein relation
• Lévy walk dynamics in an external harmonic potential
• Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling
• Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity
• Ergodic property of random diffusivity system with trapping events
• A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• Langevin picture of anomalous diffusion processes in expanding medium
• Conservative random walks in confining potentials
• Large deviations of the ballistic Lévy walk model
• The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model
• Fractional Bhatnagar-Gross-Krook kinetic equation
• Singularity of Levy walks in the lifted Pomeau-Manneville map
• Ageing in Mortal Superdiffusive Lévy Walkers
• Dynamics of a randomly kicked particle
• Strong overlap of deterministic and stochastic dynamics in a super-diffusive regime