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

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.