Quantifying the non-ergodicity of scaled Brownian motion

TL;DR

This paper analyzes the non-ergodic behavior of scaled Brownian motion with time-dependent diffusivity, providing analytical and simulation insights into ergodicity breaking across different parameters and conditions.

Contribution

It offers the first comprehensive analytical and numerical study of the ergodicity breaking parameter for scaled Brownian motion across all scaling exponents, including effects of aging.

Findings

EB parameter has no divergence at α=1/2 in long trajectories

Derived asymptotes for EB in various limits

Extended analysis to aged systems with finite observation times

Abstract

We examine the non-ergodic properties of scaled Brownian motion, a non-stationary stochastic process with a time dependent diffusivity of the form $D(t)\simeq t^{\alpha-1}$. We compute the ergodicity breaking parameter EB in the entire range of scaling exponents $\alpha$, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths $T$ and short lag times $\Delta$ the EB parameter as function of the scaling exponent $\alpha$ has no divergence at $\alpha=1/2$ and present the asymptotes for EB in different limits. We generalise the analytical and simulations results for the time averaged and ergodic properties of scaled Brownian motion in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.