Distributional Ergodicity in Stored-Energy-Driven L\'evy Flights

TL;DR

This paper introduces the stored-energy-driven Lévy flight (SEDLF), a novel random walk model where jump lengths depend on stored energy, revealing distributional ergodicity with diverse diffusive behaviors and aging effects.

Contribution

It analytically characterizes SEDLF, demonstrating its subdiffusive and superdiffusive regimes, and uncovers the intrinsic randomness and aging phenomena of diffusion coefficients.

Findings

Ensemble-averaged MSD shows subdiffusion and superdiffusion depending on coupling.

Time-averaged MSD increases linearly with time, with random diffusion coefficients.

Diffusion coefficient exhibits aging in subdiffusive regime and growth with measurement time in superdiffusive regime.

Abstract

We study a class of random walk, the stored-energy-driven L\'evy flight (SEDLF), whose jump length is determined by a stored energy during a trapped state. The SEDLF is a continuous-time random walk with jump lengths being coupled with the trapping times. It is analytically shown that the ensemble-averaged mean square displacements exhibit subdiffusion as well as superdiffusion, depending on the coupling parameter. We find that time-averaged mean square displacements increase linearly with time and the diffusion coefficients are intrinsically random, a manifestation of {\it distributional ergodicity}. The diffusion coefficient shows aging in subdiffusive regime, whereas it increases with the measurement time in superdiffusive regime.