Ergodicity breaking, ageing, and confinement in generalised diffusion processes with position and time dependent diffusivity

TL;DR

This paper investigates a generalized diffusion process with position and time-dependent diffusivity, revealing ergodicity breaking, ageing effects, and confinement impacts on diffusion characteristics.

Contribution

It introduces a unified model combining features of scaled Brownian motion and heterogeneous diffusion, with analytical results on MSD, ergodicity, and ageing.

Findings

Weak ergodicity breaking persists in the long-time limit.

Ageing effects depend on the ageing and measurement times.

Confinement significantly alters diffusion statistics.

Abstract

We study generalised anomalous diffusion processes whose diffusion coefficient $D(x,t)\sim D_0|x|^{\alpha}t^{\beta}$ depends on both the position $x$ of the test particle and the process time $t$. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalised diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalised diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.