Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

Abstract

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Great Barrier Reef degradation, sea surface temperatures, and atmospheric CO2 levels collectively exhibit a stochastic process with memory

“…The MSD, Eq. ( 7), is typical of anomalous diffusion with exponent (Safdari et al 2017;Wang et al 2020;Metzler et al 2014). Table 1 gives us the values of the memory parameter which yield, = 8.28 for the GBR, = 1.50 for CO2 levels, and = 1.36 for SST.…”

“…In particular, 0 < < 1, gives subdiffusive behavior whereas, 1 < < 2, is superdiffusive (Metzler et al 2014;Biagini et al 2008). The case = 2 is the ballistic domain, while 2 < is referred to as hyperdiffusive or hyperballistic growth of the MSD that transpires, for example, in particle diffusion in turbulent flows or hyper-transport in optical systems (Safdari et al 2017;Wang et al 2020;Peccianti and Morandotti 2012;Levi et al 2012;Siegle et al 2010;Yao et al 1995). For, 1 < , there is long range memory behavior where a fluctuating variable quickly departs from its point of origin as time evolves in comparison with ordinary Brownian motion.…”

Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

“…To obtain satisfactory and optimal models, one needs to take into account the trade-off between the simplicity and accuracy of the reduced models sought after. Indeed, one may find that a reduced model, while simplified, fails to give a physically correct model for describing a system of interest [64]. Two successful reductions were carried out in [28] for the case F 1 = 0 and in [42] for the case F 0 = 0.…”

Nonergodicity in silo unclogging: Broken and unbroken arches

“…An assortment of models more complex than the CTRW can be found in Refs. [37,38,44,[49][50][51][52][53]; they could be useful to improve the description.…”