Aging Scaled Brownian Motion

TL;DR

This paper investigates the aging properties of scaled Brownian motion (SBM), revealing complex behaviors in confined and unconfined systems, and establishing connections with continuous time random walks and ergodicity breaking.

Contribution

It provides a comprehensive analysis of aging effects in SBM, deriving mean squared displacements, first passage times, and demonstrating how aging influences ergodicity and confinement effects.

Findings

Aging factorizes with respect to lag time in SBM.

Strong aging reduces disparity between ensemble and time averages.

Confined aging SBM shows rich behaviors depending on aging time and diffusion type.

Abstract

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly non-stationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form, that is identical to the aging behavior of scale free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly non-ergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.