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Ultraslow scaled Brownian motion

  • We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . 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 a 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 agreeWe define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . 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 a 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.show moreshow less

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Metadaten
Author:Ralf MetzlerORCiDGND, Andrey G. Cherstvy, Aleksei V. ChechkinORCiDGND, Anna S. Bodrova
DOI:https://doi.org/10.1088/1367-2630/17/6/063038
ISSN:1367-2630 (online)
Parent Title (English):New journal of physics : the open-access journal for physics
Publisher:Dt. Physikalische Ges., IOP
Place of publication:Bad Honnef, London
Document Type:Article
Language:English
Date of first Publication:2015/06/29
Year of Completion:2015
Publishing Institution:Universität Potsdam
Release Date:2015/07/23
Tag:ageing; anomalous diffusion; stochastic processes
Volume:17
Issue:063038
Funder:Universität Potsdam, Publikationsfonds
Grant Number:PA 2015_12
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
PACS Classification:00.00.00 GENERAL / 05.00.00 Statistical physics, thermodynamics, and nonlinear dynamical systems (see also 02.50.-r Probability theory, stochastic processes, and statistics) / 05.60.-k Transport processes
Peer Review:Referiert
Grantor:Publikationsfonds der Universität Potsdam
Publication Way:Open Access
Licence (English):License LogoCreative Commons - Attribution 3.0 unported
Notes extern:Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 188