Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion

TL;DR

Scaled Brownian motion (SBM) is a non-stationary Gaussian process with time-dependent diffusivity, useful for modeling anomalous diffusion but fundamentally different from other models under confinement, impacting single particle tracking analysis.

Contribution

This paper clarifies the non-ergodic and non-stationary nature of SBM and highlights its limitations compared to fractional Brownian motion and continuous time random walks under confinement.

Findings

SBM is weakly non-ergodic with minimal amplitude scatter.

Under confinement, SBM differs fundamentally from other anomalous diffusion models.

SBM cannot describe particles in thermalized stationary systems.

Abstract

Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is $\langle x^2(t)\rangle\simeq\mathscr{K}(t)t$ with $\mathscr{K}(t)\simeq t^{\alpha-1}$ for $0<\alpha<2$. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used.