Ultraslow scaled Brownian motion

TL;DR

This paper introduces ultraslow scaled Brownian motion (USBM), a non-stationary stochastic process with a time-dependent diffusion coefficient, exhibiting ultraslow logarithmic MSD growth, non-ergodic behavior, and aging effects, supported by analytical and simulation results.

Contribution

The paper defines and analyzes USBM, revealing its ultraslow growth, non-ergodic nature, and aging properties, which are novel features compared to traditional scaled Brownian motion.

Findings

MSD grows logarithmically with time for unconfined USBM

MSD decays inversely in harmonic potential, showing non-stationarity

USBM exhibits weak ergodicity breaking and aging effects

Abstract

We define and study in detail \emph{utraslow scaled Brownian motion (USBM)\/} characterised by a time dependent diffusion coefficient of the form $D(t)\simeq 1/t$. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In an harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.