Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics

TL;DR

This paper investigates the ergodic properties of superdiffusive Levy walk processes, revealing finite-time effects, scatter in scaling exponents, and ultraweak ergodicity breaking, which are crucial for understanding superdiffusive dynamics.

Contribution

It introduces the analysis of finite-time effects and amplitude depression in superdiffusive Levy walks, highlighting ultraweak ergodicity breaking and the variability of scaling exponents.

Findings

Finite-time trajectories show scatter in scaling exponents around 3-alpha.

Trajectory-to-trajectory averages depend on measurement time.

Long-time averages differ from ensemble means by a constant factor.

Abstract

We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement delta**2 around the ensemble value 3-alpha (1<alpha<2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the trajectory-to-trajectory average of delta**2 as function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.